linear diophantine systems: partition analysis and
TRANSCRIPT
UNIVERSITAT LINZJOHANNES KEPLER JKU
Technisch-NaturwissenschaftlicheFakultat
Linear Diophantine Systems:Partition Analysis and Polyhedral Geometry
DISSERTATION
zur Erlangung des akademischen Grades
Doktor
im Doktoratsstudium der
Technischen Wissenschaften
Eingereicht von:
Zafeirakis Zafeirakopoulos
Angefertigt am:
Research Institute for Symbolic Computation
Beurteilung:
Prof. Peter Paule (Betreuung)Prof. Matthias Beck
Linz, December, 2012
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This version of the thesis was updated in October 2014. The update only concernsthe presentation and not the content of the thesis.
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Abstract
The main topic of this thesis is the algorithmic treatment of two problems relatedto linear Diophantine systems. Namely, the first one is counting and the second one islisting the non-negative integer solutions of a linear system of equations/inequalities.
The general problem of solving polynomial Diophantine equations does not admit analgorithmic solution. In this thesis we restrict to linear systems, so that we can treatthem algorithmically. In 1915, MacMahon, in his seminal work โCombinatory Anal-ysisโ, introduced a method called partition analysis in order to attack combinatorialproblems subject to linear Diophantine systems. Following that line, in the beginningof this century Andrews, Paule and Riese published fully algorithmic versions of par-tition analysis, powered by symbolic computation. Parallel to that, the last decadessaw significant progress in the geometric theory of lattice-point enumeration, startingwith Ehrhart in the 60โs, leading to important theoretical results concerning generatingfunctions of the lattice points in polytopes and polyhedra. On the algorithmic side ofpolyhedral geometry, Barvinok developed the first polynomial-time algorithm in fixeddimension able to count lattice points in polytopes.
The main goal of the thesis is to connect these two lines of research (partition analysisand polyhedral geometry) and combine tools from both sides in order to construct betteralgorithms.
The first part of the thesis is an overview of conic semigroups from both a geometricand an algebraic viewpoint. This gives a connection between polyhedra, cones andtheir generating functions to generating functions of solutions of linear Diophantinesystems. Next, we provide a geometric interpretation of Elliott Reduction and Omega2(two implementations of partition analysis). The partial-fraction decompositions used inthese two partition-analysis methods are interpreted as decompositions of cones. Withthis insight and employing tools from polyhedral geometry, such as Brionโs theoremand Barvinokโs algorithm, we propose a new algorithm for the evaluation of the ฮฉโฅoperator, the central tool in MacMahonโs work on partition analysis. This gives animplementation of partition analysis heavily based on the geometric understanding ofthe method. Finally, a classification of linear Diophantine systems is given, in order tosystematically treat the algorithmic solution of linear Diophantine systems, especiallythe difference between parametric and non-parametric problems.
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Zusammenfassung
Das Hauptaugenmerk dieser Arbeit liegt auf der algorithmischen Losung zweier Prob-leme, welche im Zusammenhang mit linearen diophantischen Systemen auftreten. Daserste Problem ist die Bestimmung der Anzahl der nicht-negativen, ganzzahligen Losun-gen linearer Systeme von (Un-)Gleichungen, das zweite die Auflistung ebendieser Losun-gen.
Die Losung von polynomiellen diophantischen Gleichungen ist nicht ohne Einschran-kungen algorithmisch handhabbar. Aus diesem Grund beschranken wir uns in der vor-liegenden Arbeit auf lineare Systeme. 1915 fuhrte MacMahon in seinem Werk โCom-binatory Analysisโ eine Methode namens Partition Analysis ein, um kombinatorischeProbleme zu losen, in denen lineare diophantische Systeme auftreten. Darauf aufbauend,jedoch unter weitreichender Miteinbeziehung von Techniken des symbolischen Rechnens,veroffentlichten Andrews, Paule und Riese zu Beginn dieses Jahrhunderts algorithmischeVarianten der Partition Analysis. Parallel dazu gab es in den letzten Jahrzehnten er-hebliche Fortschritte in der geometrischen Theorie zur Aufzahlung von Gittervektoren,beginnend mit Erhart in den 70er Jahren, welche zu wichtigen theoretischen Erkenntnis-sen bezuglich erzeugender Funktionen von Gittervektoren in Polytopen und Polyedernfuhrten. Einen wesentlichen Beitrag in der algorithmischen Entwicklung leistete Barvi-nok mit der Formulierung des ersten Algorithmus, der die Aufzahlung von Gittervektorenin Polytopen ermoglicht und bei fester Dimension eine polynomielle Laufzeit aufweist.
Das Ziel dieser Arbeit is es, beide Forschungsstrange -Partition Analysis und PolyederGeometrie- zu verbinden und die Werkzeuge aus beiden Theorien zu neuen, besserenAlgorithmen zu kombinieren. Der erste Teil gibt einen Uberblick uber konische Halb-gruppen aus zweierlei Blickwinkeln, dem geometrischen und dem algebraischen. Diesermoglicht es uns, die Verbindung von Polyedern, Kegeln und ihrer erzeugenden Funk-tionen zu den erzeugenden Funktionen der Losungen von linearen diophantischen Sys-temen herzustellen. Darauf folgend behandeln wir geometrische Interpretationen vonElliotts Reduction und Omega2, zweier Realisierungen von Partition Analysis. Die Par-tialbruchzerlegungen, welche in diesen beiden Methoden auftreten, werden als Zerlegun-gen von Kegeln interpretiert. Zusammen mit geeigneten Hilfsmitteln aus der PolyederGeometrie, wie etwa dem Satz von Brion und Barvinoks Algorithmus, fuhrt dies zurFormulierung eines neuen Algorithmus zur Anwendung des Omega-Operators, dem zen-tralen Werkzeug in MacMahons Arbeit an Partition Analysis. Wir erhalten darauseine algorithmische Realisierung von Partition Analysis, die entscheidend auf dem ge-ometrischen Verstandnis der Methode basiert. Abschlieรend wird eine Klassifikationvon linearen diophantischen Systemen eingefuhrt, um systematisch deren algorithmischeLosung aufarbeiten zu konnen. Speziell wird dabei auf die Unterscheidung zwischenparametrischen und nicht-parametrischen Problemen geachtet.
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Ich erklare an Eides statt, dass ich die vorliegende Dissertation selbststndig und ohnefremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutztbzw. die wortlich oder sinngemaร entnommenen Stellen als solche kenntlich gemachthabe. Die vorliegende Dissertation ist mit dem elektronisch bermittelten Textdokumentidentisch.
Zafeirakis Zafeirakopoulos
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Acknowledgments
Writing a thesis proved to be harder than what I thought. But, now I know, it wouldhave been impossible without the help of the people who supported me during the last3-4 years.
First I want to thank my advisor Prof. Peter Paule for his support in a range of issuesspanning scientific and everyday life. The two most important things I am grateful to himfor are his suggestion for my PhD topic and his support for my participation in the AIMSQuaRE for โPartition Theory and Polyhedral Geometryโ. When I first mentioned tohim my interests, Prof. Paule successfully predicted that the topic of this thesis would befor me exciting enough so that even passing through frustrating and exhausting periods,today I am still as interested as the first day.
Concerning the SQuaRE meetings in the American Institute of Mathematics, I can-not exaggerate about how helpful they were for me. The three weeks of these meetingswere undoubtedly among the most intensive learning and working experiences I ever had.Each one of the participants - Prof. Matthias Beck, Prof. Ben Brown, Prof. MatthiasKoppe and Prof. Carla Savage - helped me in a different way to understand parts of thebeauty lying in the intersection of algebraic combinatorics, polyhedral geometry, numbertheory and the algorithmic methods involved.
Prof. Beck acted as a host during my five months stay in San Francisco visiting SanFrancisco State University in the context of a Marshall Plan Scholarship. Except forbeing excellent in his formal duty as a host (advising me during that period), Matt andTendai acted as real hosts, making my stay in San Francisco a very positive experienceboth in academic and social terms.
During my stay in San Francisco, I met Dr. Felix Breuer with whom I spent agood part of my time there discussing about interesting topics, not always concerningmathematics. During our mathematical discussions emerged a number of topics thathopefully I will have the pleasure to pursue in the future, as well as one chapter of thisthesis. Landing in a city as a stranger always gives a strange feeling, but Felix andChambui contributed in making this feeling negligible.
The first time I landed in a new city was in 2009. The task of making this moveeasier fell on the hands of Veronika, Flavia, Burcin and Ionela. To a large extend, I owethe fact that I am here to them and the great job they did in making Linz a warm place(which is not easy in any sense, especially after living 27 years in Athens), when I firstarrived.
PhD studies is like an extended summer camp. And I donโt just mean exhaustingbut fun. You meet people you like and then they leave, or you leave. You do not knowwhen you will see each other again, possibly next summer in a conference. But, if youare fortunate enough, you make some friends. I was fortunate to meet Max, Hamid,Jakob, Cevo, Nebiye and Madalina, who I do consider as friends.
Distance makes communication between friends harder, but you know that you careabout them and you hope they do the same. Although I love Athens as a city, aftertaking the time to think about what I really miss, it is the people. It is the friends I
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know for half my life. A couple of them for more than that. For me, the most frustratingthing about living abroad was not being there in the little moments you never care about,except if you are not there.
Giorgos, Dimitris and NN, being themselves in a position similar to mine, distractedme efficiently through these years with pointless but enjoyable discussions. Elias provedto be always a valuable source of information and support. I also want to thank mycolleagues at RISC and especially Manuel, Matteo, Ralf and Veronika for the very usefuldiscussions and directions they gave me on various mathematical topics. At this pointI can say that the bridge at RISC is the place where I made the most friends and alsothe most work done these last years.
Finally, I want to thank my family. My parents supported me in all possible waysduring my studies in Greece and keep supporting me until today. I do owe them morethan I can possibly offer. My brother is still in Athens, in a stay-behind role. I owe tohim my luxury of being able to leave in order to pursue my goals. Achieving one goalonly asks for the next one. And for me that is to be able to support Ilke for the restof my life, the way she supported me during these years. Good that she is an endlesssource of happiness and inspiration for me.
Contents
1 Geometry, Algebra and Generating Functions 151.1 Polyhedra, Cones and Semigroups . . . . . . . . . . . . . . . . . . . . . . 151.2 Formal Power Series and Generating Functions . . . . . . . . . . . . . . . 301.3 Formal Series, Rational Functions and Geometry . . . . . . . . . . . . . . 41
2 Linear Diophantine Systems 512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Partition Analysis 633.1 Partition Analysis Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Algebraic Partition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Partition Analysis via Polyhedral Geometry 914.1 Eliminating one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2 Eliminating multiple variables . . . . . . . . . . . . . . . . . . . . . . . . . 984.3 Improvements based on geometry . . . . . . . . . . . . . . . . . . . . . . . 1014.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A Proof of โGeometry of Omega2โ 111
9
Introduction
Some History
One of the earliest recorded uses of geometry as a counting tool is the notion of figuratenumbers. Ancient Greeks used pebbles (ฯฮฑฮปฮฏฮบฮน, where the word calculus comes from),in order to do arithmetic. For example, an arrangement of pebbles would be used tocalculate triangular numbers. In general figurate numbers were popular in ancient times.
Today we mostly care about squares and some special cases (like pentagonal numbersin Eulerโs celebrated theorem). Among the mathematicians that were interested infigurate numbers was, of course, Diophantus who wrote a book on polygonal numbers(see [25]). After an interesting turn of events, one of the most prominent methods forthe solution of linear Diophantine systems relies on generalizing figurate numbers.
This generalization bears the name of Eugene Ehrhart who made the first major con-tributions to what is today called Ehrhart theory. The goal is to compute and investigateproperties of functions enumerating the lattice points in polytopes. In dimension 2 andfor the simple regular polygons this coincides with the concept of figurate numbers, butnaturally we are interested in higher-dimensional polytopes and polytopes with morecomplicated geometry.
Diophantus in his masterpiece โArithmeticaโ dealt with the solution of equations[25]. But in the time of Diophantus, a couple of things were essentially different than inmodern mathematics:
โ No notion of zero existed.
โ Fractions were not treated as numbers (Diophantus was the first to do so).
โ There was no notation for arithmetic.
โArithmeticaโ consisted of 13 books, out of which only six survived and possibly an-other four through Arab translations (found recently), dealing with the solution of 130equations. On one hand his work is important because it is the oldest account we havefor indefinite equations (equations with more than one solutions). More importantlythough, Diophantus introduced a primitive notational system for (what later was called)algebra.
For our purposes, the essential part of his work is his view on what is the solution ofan equation. He considered equations with positive rational coefficients whose solutions
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12 CONTENTS
are positive rationals. Following this path, we consider a Diophantine problem to haveinteger coefficients and non-negative integer solutions.
In 1463 the German mathematician Regiomontanus, traveling to Italy, he came acrossa copy of Arithmetica. He considered it to be an important work when he reported to afriend of his about it. He intended to do the translation, but he could only find six outof the thirteen books that Diophantus mentioned in his introduction, thus he did notproceed [14].
Although the most famous marginal note to be found in a copy of Arithmeticais by Fermat (his last theorem), there is another one which is very interesting. TheByzantine scholar Chortasmenos notes โThy soul, Diophantus, be with Satan becauseof the difficulty of your problemsโ (funny enough next to what came to be known asFermatโs last theorem).
This last comment, combined with the note that Diophantus did not have a gen-eral method (after solving the 100th problem, you still have no clue how to attack the101st) is important for us. Of course, for non-linear Diophantine problems one cannotexpect a general method (Hilbertโs 10th problem). But we present algorithmic solutions(developed during the last century) for linear Diophantine systems and examine theirconnections.
Contributions
The main contribution of this thesis is in the direction of connecting partition analysis, amethod for solving linear Diophantine systems, with polyhedral geometry. The analyticviewpoint of traditional partition analysis methods (due to Elliott, MacMahon, An-drews, Paule and Riese) is interpreted in the context of polyhedral geometry. Throughthis interpretation we are allowed to use tools from geometry in order to enhance thealgorithmic procedures and the understanding of partition analysis. More precisely, themain contributions are:
Geometric and algebraic interpretation of partition analysis
โ In Theorem 3.2 a geometric interpretation of the generalized partial fraction de-composition employed in Omega2 is given.
โ We define geometric objects related to the crude generating function and providea geometric version of the ฮฉโฅ operator.
โ An algebraic version of ฮฉโฅ is given based on gradings of algebraic structures anda variant of Hilbert series.
CONTENTS 13
Algorithmic contributions to partition analysis
โ A new algorithm for the evaluation of the ฮฉโฅ operator, following the traditionalparadigm of recursive elimination of ๐ variables (see Section 4.1).
โ A new algorithm, motivated by the geometric understanding of the action of ฮฉโฅ,that performs simultaneous elimination of all ๐ variables (see Section 4.2).
A second goal of the thesis, especially given the interest in algorithmic solutions, isthe classification of linear Diophantine problems. In the literature the notion of a linearDiophantine problem is not consistent and depends on the motivation of the author.
Finally, we note that the main reason why we believe it is important to explore suchconnections, although in both contexts (partition analysis and polyhedral geometry)there are already methods to attack the problem, is that knowledge transfer from onearea to another often helps to develop algorithmic tools that are more efficient.
Chapter 1
Geometry, Algebra andGenerating Functions
In this chapter we present a basic introduction to geometric, algebraic and generatingfunction related concepts that will be used in later chapters. This introduction is notmeant to be complete but rather a recall of basic notions mostly to set up notation. InSection 1.1, some geometric notions and the discrete analog of cones are presented. Fora detailed introduction see [17, 11, 38]. In Section 1.2 a hierarchy of algebraic structuresrelated to formal power series is presented first and then an introduction to generatingfunctions is given. Finally, in Section 1.3 we present the relation of formal power seriesand generating functions via polyhedral geometry and give a short description of moreadvanced geometric tools we will use later.
We note that in this chapter we will use boldface fonts for vectors and the multi-indexnotation x๐ = ๐ฅ๐11 ๐ฅ๐22 . . . , ๐ฅ๐๐๐ .
1.1 Polyhedra, Cones and Semigroups
In this section we give a short introduction to polyhedra and polytopes. We define thefundamental objects of polyhedral geometry and provide terminology and notation thatwill be used later. For a more detailed introduction to polyhedral geometry see [17, 38]and references therein.
All of the theory developed in this section takes place in some Euclidean space. Forsimplicity of notation and clarity, we agree this to be R๐ for some ๐ โ N. We fix ๐ todenote the dimension of the ambient space R๐.
In Section 1.1.1 we will introduce polyhedra and in Section 1.1.2 we discuss aboutpolytopes, while in Section 1.1.3 we will present definitions and notation related topolyhedral cones. In Section 1.1.4 we introduce semigroups and their connection topolyhedral cones.
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16 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
1.1.1 Polyhedra
There are many different ways to introduce the notion of polyhedra in literature [11, 17,38, 12]. According to our intuition, polyhedra are geometric objects with โflat sidesโ.Thus, it is expected that we will resort to some sort of linear relations defining polyhedra.In order to make this more precise we will use linear functionals.
Definition 1.1 (Linear Functional)Given a real vector space R๐, a linear functional is a map ๐ : R๐ โ R such that
๐(๐๐ข+ ๐๐ฃ) = ๐๐(๐ข) + ๐๐(๐ฃ)
for all ๐ข, ๐ฃ โ R๐ and ๐, ๐ โ R. ๏ฟฝ
Observe that any linear equation/inequality in ๐ variables ๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐, i.e.
๐โ๐=1
๐๐๐ฅ๐ โฅ ๐ for ๐1, ๐2, . . . , ๐๐, ๐ โ R
can be expressed as ๐(x) โฅ ๐ in terms of a linear functional
๐(x) = (๐1, ๐2, . . . , ๐๐)๐x.
For a โ R๐, we denote by ๐๐(x) the linear functional ๐๐(x) = (๐1, ๐2, . . . , ๐๐)๐x.
Having a concise and formal way to express linear relations, the next step is to definethe solution space of a linear equation/inequality. Given a โ R๐ and ๐ โ R, the solutionspace of ๐๐(x) = ๐ is called an affine hyperplane and is denoted by ๐ป๐,๐, i.e.,
๐ป๐,๐ = {x โ R๐ : ๐๐(x) = ๐}.
On the other hand, given a โ R๐ and ๐ โ R, the solution space of the inequality๐๐(x) โฅ ๐ is called an affine halfspace and is denoted by โ+
๐,๐, i.e.,
โ+๐,๐ = {x โ R๐ : ๐๐(x) โฅ ๐} for some ๐ โ R.
Naturally, a second half-space is given by โโ๐,๐ = {x โ R๐ : ๐๐(x) โค ๐}. We note that the
hyperplane ๐ป๐,๐ is the intersection of the two halfspaces โ+๐,๐ and โ
โ๐,๐, or, equivalently,
the hyperplane divides the Euclidean space into two halfspaces.
1.1. POLYHEDRA, CONES AND SEMIGROUPS 17
๐ฅ
๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
A hyperplane and a halfspace it defines.
Letโs now consider (finite) intersections of halfspaces. Given two linear functionals ๐๐1and ๐๐2 for some a1,a2 โ R๐, and two scalars ๐1, ๐2 โ R, we consider the intersection ofthe two halfspaces
โ+๐1,๐1โฉโ+
๐2,๐2={x โ R๐ : ๐๐1(x) โฅ ๐1, ๐๐2(x) โฅ ๐2
}For simplicity, we omit the linear functional notation and use matrix notation, as itis customary for systems of linear inequalities. In other words, given a set of vectorsai โ R๐ and scalars ๐๐ โ R for ๐ โ [๐], we writeโ
๐โ[๐]
โ+๐๐,๐๐
={x โ R๐ : ๐ดx โฅ ๐
}
where ๐ด is the matrix with ai as its ๐-th row and ๐ = (๐1, ๐2, . . . , ๐๐)๐ .
With the above notation and terminology we can proceed with the definition of the firstfundamental object in polyhedral geometry.
Definition 1.2 (Polyhedron)A polyhedron is the intersection of finitely many affine halfspaces in R๐. More precisely,given ๐ด โ R๐ร๐ and b โ R๐, then the polyhedron ๐ซ๐ด,๐ is the subset of R๐
๐ซ๐ด,๐ = {x โ R๐ : ๐ดx โฅ ๐}.
๏ฟฝ
An important subclass of polyhedra, in which we will restrict for the rest of the thesis, isthat of rational polyhedra. In the definition of ๐ซ๐ด,๐ we assume ๐ด โ R๐ร๐ and b โ R๐.If ๐ดโฒ โ Z๐ร๐ and ๐โฒ โ Z๐ can be chosen such that ๐ซ๐ด,๐ = ๐ซ๐ดโฒ,๐โฒ , then the polyhedron๐ซ๐ด,๐ is called rational. For brevity, if no characterization is given, then we assume apolyhedron is rational.
18 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
๐ฅ
๐ฆ
๐ง
A polyhedron
In the beginning of the section we fixed ๐ to denote the dimension of the ambientEuclidean space. Moreover, in the definitions of halfspaces and polyhedra we assumedthat these are subsets of R๐. Nevertheless, the dimension of a polyhedron is not neces-sarily equal to the dimension of the ambient space as we shall see. In order to define thenotion of the dimension of a polyhedron, we will use the affine hull of a set. Given a realvector space R๐ and a set ๐ โ R๐, then affhullR๐ ๐ is the set of all affine combinationsof elements in ๐, i.e.,
affhullR๐ ๐ =
{๐โ
๐=1
๐ผ๐si
๐ โ N*,
๐โ๐=1
๐ผ๐ = 1, ๐๐ โ R
}.
Let ๐ซ๐ด,๐ be a polyhedron in the ambient Euclidean space R๐. The dimension of๐ซ๐ด,๐, denoted by dim๐ซ๐ด,๐ is defined to be the dimension of the affine hull of ๐ซ๐ด,๐, i.e.,
dim๐ซ๐ด,๐ = dimR๐ (affhullR๐ ๐ซ๐ด,๐) .
Observe that dim๐ซ๐ด,๐ โค ๐ in general. If dim๐ซ๐ด,๐ = ๐, then we say that ๐ซ๐ด,๐ is fulldimensional. A ๐-polyhedron is a polyhedron of dimension ๐.
1.1. POLYHEDRA, CONES AND SEMIGROUPS 19
๐ฅ
๐ฆ
๐ง
A non full-dimensional polyhedron. The ambient space dimension is 3, while thedimension of the polyhedron is 2.
1.1.2 Polytopes
Although we will mostly deal with polyhedra, a very important object, especially whenit comes to lattice-point enumeration, is the polytope. Among the numerous equivalentdefinitions of a polytope, we chose the following:
Definition 1.3 (Polytope)A bounded polyhedron is called a polytope. ๏ฟฝ
A 3-polytope.
One of the fundamental properties of polytopes, which we will also use in laterchapters, is that a polytope has two equivalent representations, the H-representationand the V-representation. H-representation stands for halfspace representation and itis the set of halfspaces whose intersection is the polytope, i.e., the description we usedin the definition. V-representation stands for vertex representation. A polytope is theconvex hull of its vertices, thus it can be described by a finite set of points in R๐. For adetailed proof of this non-trivial fact see Appendix A in [17].
20 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
We note that the convex hull of a set of ๐ points ๐ โ R๐ is the smallest convex setcontaining ๐, or alternatively, the set of all convex combinations of elements from ๐, i.e.,
{๐โ
๐=1
๐ผ๐si
si โ ๐,
๐โ๐=1
๐ผ๐ = 1, ๐ผ โ R
}.
In order to establish terminology and to define the notion of vertex used above, wedefine the notions of supporting hyperplanes and faces of polyhedra and polytopes. Ahyperplane ๐ป โ R๐ is said to support a set ๐ โ R๐ if
โ ๐ is contained in one of the two closed halfspaces determined by ๐ป,
โ there exists ๐ฅ โ ๐ such that ๐ฅ โ ๐ป.
In other words, a supporting hyperplane ๐ป for the polyhedron ๐ is a hyperplane thatintersects ๐ and leaves ๐ entirely on one side.
๐ฅ๐ฆ
z
๐ฅ๐ฆ
๐ง
Supporting hyperplanes for a cube in 3D
With the use of supporting hyperplanes, we can define now the faces of a polyhedron orpolytope ๐ in R๐. A face ๐น of ๐ is the intersection of ๐ with a supporting hyperplane๐. The dimension of a face ๐น is the dimension of affhullR๐ ๐น . A face of dimension๐ is called a ๐-face. In particular a 0-face is called vertex, a 1-face is called edge orextreme ray and a (dim๐ โ 1)-face is called facet. We note that a face of a polyhedron(resp. polytope) is again a polyhedron (resp. polytope) and that the definition of facedimension is compatible with the the definition of the dimension of a polyhedron.
1.1. POLYHEDRA, CONES AND SEMIGROUPS 21
A facet, a 1-face and a vertex of a 3-polytope.
Simplices are polytopes of very simple structure and are used as building blocks inpolyhedral geometry. We first define the standard simplex in dimension ๐.
Definition 1.4 (Standard Simplex)The standard ๐-simplex is the subset of R๐ defined by
ฮ๐ =
{(๐ฅ1, ๐ฅ2 . . . , ๐ฅ๐) โ R๐
๐โ๐=1
๐ฅ๐ โค 1 and ๐ฅ๐ โฅ 0 โ๐
}Equivalently, the standard ๐-simplex ฮ๐ is the convex hull of the ๐+ 1 points:
๐0 = (0, 0, 0, . . . , 0), ๐1 = (1, 0, 0, . . . , 0), ๐2 = (0, 1, 0, . . . , 0), . . . ๐๐ = (0, 0, 0 . . . , 1).
๏ฟฝ
In general a ๐-simplex is defined to be the convex hull of ๐ + 1 affinely independentpoints in R๐.
๐ฅ๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
The standard 3-simplex and the simplex defined by (1, 1, 0), (2, 3, 0), (0, 3, 0) and(1, 2, 4).
22 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
1.1.3 Cones
In polyhedral geometry, the main object is the polyhedron but the main tool is thepolyhedral cone. It is usual to use cones in order to compute with polyhedra andpolytopes or in general to investigate their properties. A fundamental relation is givenby Brionโs theorem (see Theorem 1.3), connecting polyhedra and cones. Cones arepolyhedra of a special type. They are finite intersections of linear halfspaces.
Definition 1.5 (Polyhedral Cone)Given a set of linear functionals ๐๐(x) : R๐ โฆโ R for ๐ โ [๐], we define the polyhedralcone
๐ถ ={x โ R๐
๐๐(x) โฅ 0 for all ๐ โ [๐]
}๏ฟฝ
For the definition of a cone we used inequalities, complying with the definition of poly-hedron. In what follows though, we will most often define cones via their generators.We first define the conic hull of a set of vectors.
Definition 1.6Given a set of vectors {v1,v2, . . . ,vk} โ R๐, its conic hull is
co(v1,v2, . . . ,vk) =
{๐โ
๐=1
๐๐vi
๐๐ โ R, ๐๐ โฅ 0
}.
๏ฟฝ
The following lemma certifies that a cone can be defined via a finite set of generators.
Lemma 1. A polyhedral cone is the conic hull of its (finitely many) extreme rays. ๏ฟฝ
Now, given a1,a2, . . . ,ak โ R๐, we define the cone generated by a1,a2, . . . ,ak as
๐R (a1,a2, . . . ,ak) = co (a1,a2, . . . ,ak) =
{x โ R๐
๐ฅ =
๐โ๐=1
โ๐ai, โ๐ โฅ 0, โ๐ โ R
}.
A cone will be called rational if there exists a set of generators for the cone such thatthe generators contain only rational coordinates. As with polyhedra, since all our conesare rational, we omit the characterization.
1.1. POLYHEDRA, CONES AND SEMIGROUPS 23
๐ฅ
๐ฆ
๐ง
A polyhedral cone.
Among cones, there are two special types that have a central role in computationalpolyhedral geometry. These are simplicial and unimodular cones. A cone in R๐ is calledsimplicial if and only if it is generated by linearly independent vectors in R๐. We notethat although usually the definition of a simplicial cone asserts ๐ linearly independentgenerators, we do not enforce this restriction. The reason is that often our cones, andpolyhedra in general, will not be full dimensional.
A simplicial cone.
The connection to simplices is more than nominal, in particular a section by a hyperplane(not containing the cone apex) of a simplicial ๐-cone is a (๐ โ 1)-simplex. For the restof the section, except if stated differently, our cones are simplicial.
An object encoding all the information contained in a cone is the fundamentalparallelepiped of the cone.
Definition 1.7 (Fundamental Parallelepiped)Given a simplicial cone ๐ถ = ๐R (a1,a2, . . . ,ad) โ R๐, we define its fundamental paral-lelepiped as
ฮ R(๐ถ) =
{๐โ
๐=1
๐๐ai
๐๐ โ [0, 1)
}
24 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
and the set of lattice points in the fundamental parallelepiped as
ฮ Z๐(๐ถ) = ฮ R(๐ถ) โฉ Z๐
๏ฟฝ
Note that we assume as working lattice the standard lattice Z๐.In the figure below we can see that the cone is spanned by copies of its fundamental
parallelepiped. This is a general fact, i.e., any simplicial cone is spanned by copies of itsfundamental parallelepiped.
(1, 0)
(1, 3)
+(1, 3)
+(1, 0)
The fundamental parallelepiped of the cone generated by (1, 0) and (1, 3).
A cone ๐ถ is called unimodular if and only if ฮ Z๐(๐ถ) = {0}. An equivalent definitionis that the matrix of cone generators is a unimodular matrix, i.e., it has determinantequal to ยฑ1. The following lemma gives a condition for a non full-dimensional cone tobe unimodular.
Lemma 2. Let ๐ถ = ๐R (a1,a2, . . . ,an) in R๐, for ๐ โค ๐. Let ๐บ = [a1,a2, . . . ,an],the matrix with ai as columns. If ๐บ contains a full rank square ๐ ร ๐ submatrix withdeterminant ยฑ1, then ๐ถ is unimodular. ๏ฟฝ
Proof. Let ๐ผ be an index set such that [๐บ๐]๐โ๐ผ is a full rank square ๐ร๐ submatrix withdeterminant ยฑ1, where ๐บ๐ is the ๐-th row of ๐บ. Without loss of generality rearrange thecoordinate system such that ๐ผ are the first ๐ coordinates. Let ๐ถ๐ be the (orthogonal)projection of ๐ถ into ๐ ๐. Then ๐ถ๐ is unimodular. Since projection maps lattice pointsto lattice points, the only way that ๐ถ is not unimodular is that ฮ (๐ถ) contains latticepoints that all project to the origin. In that case, there would be a generator of theform (0, 0, . . . , 0,an+1,an+2, . . . ,am), which contradicts the assumption that [๐บ๐]๐โ๐ผ isfull rank.
All cones we have seen until now are pointed cones, i.e., they contain a vertex.According to the definition of cone, this is not necessary. A cone may contain a line orhigher dimensional spaces. It is easy to see though, that a cone is pointed if and onlyif it does not contain a line. The unique vertex of a pointed cone is called the apex ofthe cone. Observe that simplicial cones are always pointed.
1.1. POLYHEDRA, CONES AND SEMIGROUPS 25
1.1.4 Semigroups
A semigroup is a very basic algebraic structure, essentially expressing that a set has awell behaved operation. The semigroup operation will be called addition and will bedenoted by +.
Definition 1.8 (Semigroup)A semigroup is a set ๐ together with a binary operation + such that
โ For all ๐, ๐ โ ๐ we have ๐+ ๐ โ ๐.
โ For all ๐, ๐, ๐ โ ๐, we have (๐+ ๐) + ๐ = ๐+ (๐+ ๐).
๏ฟฝ
If a semigroup has an identity element, i.e., there exists ๐ in ๐ such that for all ๐ in๐ we have ๐ + ๐ = ๐ + ๐ = ๐, then it is called a monoid. The standard example of amonoid is the set of natural numbers N.
Let ๐ be a semigroup with operation +. A set ๐ด โ ๐ is called a subsemigroup of ๐if it is closed under the operation +, i.e., for all ๐ and ๐ in ๐ด we have that ๐+ ๐ is in ๐ด.
An important notion connecting the algebraic and geometric points of view is thatof an affine semigroup. We note that all semigroups we consider in what follows areaffine (except if the contrary is explicitly stated) due to Theorem 1.1. For brevity wemay omit the characterization affine later.
Definition 1.9 (Affine semigroup [20])A finitely generated semigroup that is isomorphic to a subsemigroup of Z๐ for some ๐ iscalled affine semigroup. We will call rank(๐) or dimension of ๐ the number ๐, i.e., theleast dimension such that the semigroup ๐ is a subsemigroup of Z๐. ๏ฟฝ
The following theorem by Gordan, in the frame of invariant theory, shows the con-nection between cones and semigroups.
Theorem 1.1 (Gordanโs Lemma, [33])If ๐ถ โ R๐ is a rational cone and๐บ a subgroup of Z๐, then ๐ถโฉ๐บ is an affine semigroup. ๏ฟฝ
Gordanโs Lemma allows us to think of lattice points in cones as semigroups. We define,the other way around, the cone of a semigroup.
Definition 1.10 (Cone of ๐)Given a subsemigroup ๐ of Z๐, let ๐ (๐) be the smallest cone in R๐ containing ๐. ๏ฟฝ
Given a semigroup ๐, a set {๐1, ๐2, . . . , ๐๐} โ ๐ is called a generating set for ๐ if andonly if for all ๐ in ๐ there exist โ1, โ2, . . . , โ๐ in N such that ๐ =
โ๐๐=0 โ๐๐๐. We say that
the semigroup ๐ is generated by ๐บ โ ๐ if ๐บ is a generating set for ๐. Note that we donot set any kind of requirements about the elements of the generating set.
We already used the term lattice points in an intuitive way. Formally lattice pointsare elements of a lattice and lattice is a group isomorphic to Z๐ (see [20]]). If no otherlattice is specified, then one should assume that lattice means Z๐.
We next investigate semigroups that have special structural properties. Let ๐ bethe intersection of a rational cone with a subgroup of Z๐ (due to Gordanโs Lemma the
26 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
intersection is a semigroup). An important question is whether the semigroup generatedby the generators of the cone is equal to ๐. In order to formalize the question we needthe notions of integral element and integral closure.
Definition 1.11 (Integral element, [20])Given a lattice โ and a subsemigroup ๐ of โ, an element ๐ฅ โ โ is called integral over ๐if ๐๐ฅ โ ๐ for some ๐ โ N. ๏ฟฝ
Definition 1.12 (Integral Closure)Given a lattice โ and a semigroup ๐ of โ, the set of all elements of โ that are integralover ๐ is called the integral closure of ๐ in โ and is denoted by ๐๐ฟ. ๏ฟฝ
A semigroup ๐ is called integrally closed over (or saturated in) a lattice โ if ๐ = ๐โ .Thus, the question above becomes โis ๐ saturated?โ. When we say that ๐ is saturated,without specifying the lattice, we mean that it is saturated in Z๐. Note that most authorswould mean that the semigroup is saturated in its group of differences Z๐, rather thanZ๐.
The following proposition provides the connection between a saturated semigroupand its cone.
Proposition 1 (2.1.1. in [20]). Let ๐ be an affine semigroup of the lattice โ generatedby ๐1, ๐2, . . . , ๐๐. Then
โ ๐โ = โ โฉ ๐R (๐1, ๐2, . . . , ๐๐)
โ โ๐ 1, ๐ 2, . . . , ๐ ๐ such that ๐โ =โ๐
๐=1 ๐ ๐ + ๐
โ ๐โ is an affine semigroup.
๏ฟฝ
Now it is clear that there are three objects we could think of as cones, namely thepolyhedral cone, the lattice points contained in the polyhedral cone and the semigroupgenerated by the generators of the cone. This already shows that we need some notationto distinguish these cases, but before establishing it, we will examine cones whose apicesare not the origin and cones that have open faces.
Let ๐ถ โ R๐ be the cone ๐R (๐1, ๐2, . . . , ๐๐) and let ๐ถ โฒ โ R๐ be the cone ๐ถ translatedby the vector ๐ โ R๐. Then we have a bijection ๐ : R๐ โ R๐ given by ๐(๐ฅ) = ๐ + ๐ฅ,where addition is the vector space addition, such that ๐ถ โฒ = ๐(๐ถ). As long as ๐ is in thelattice โ, then lattice points are mapped to lattice points. Although ๐ is a semigrouphomomorphism, it is not a monoid homomorphism. This is why we prefer to view themonoids coming from cones with apex at the origin as semigroups and not monoids. It isimportant to note that any unimodular transformation, i.e., a linear transformation givenby a square matrix with determinant ยฑ1, preserves the lattice point structure. 1 Thus
1for translation we have to homogenize in order to have a square matrix with determinant ยฑ1 givingthe transformation. Otherwise we can allow transformations given by a unimodular matrix plus atranslation.
1.1. POLYHEDRA, CONES AND SEMIGROUPS 27
there exists a bijection between the lattice points of the original and the transformedcone. This fact will be used later in order to enumerate lattice points in the fundamentalparallelepiped of a cone.
Translation of cones.
Assume ๐ถ is a simplicial cone and let [๐] = ๐ผ โ N be the index set for its generators.Then each face of ๐ถ can be identified with a set ๐ผ๐น โ ๐ผ, since a face has the form{
๐ฅ โ R๐
๐ฅ =
๐โ๐=1
โ๐๐๐, โ๐ โ R, โ๐ > 0 for ๐ /โ ๐ผ๐น , โ๐ = 0 for ๐ โ ๐ผ๐น
}.
Essentially, this means that the face is generated by a subset of the generators. Notethat ๐ผ๐น contains the indices of the generators of the simplicial cone ๐ถ that are not usedto generate ๐น . A facet is identified with the index of the single generator not used togenerate it.
๐ฅ
๐ฆ
๐1
๐2
๐3{2}
{1, 2}
Half-open cones
Given a cone ๐ถ, removing from ๐ถ the points lying on a facet ๐น of ๐ถ, we obtaina cone ๐ถ โฒ that has the same faces as ๐ถ except for the faces included in ๐น . From the
28 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
previous discussion, for each facet we need only to mention the generator correspondingto that facet. We will use a 0 โ 1 vector ๐ to express the openess of a cone, i.e., theopeness vector has an entry of 1 in the position ๐ if the facet corresponding to the ๐-thgenerator is open. If ๐ผ = [๐] then the cone is open and if โ = ๐ผ โ [๐] then the cone ishalf-open.
The above discussion motivates an updated definition of polyhedral cones and relatednotation.
Definition 1.13Given ๐1, ๐2, . . . , ๐๐, ๐ โ Z๐ and an index set ๐ผ โ [๐], we define:
โ the (real) cone generated by ๐1, ๐2, . . . , ๐๐ at ๐ as
๐๐ผR (๐1, ๐2, . . . , ๐๐; ๐) =
{๐ฅ โ R๐ : ๐ฅ = ๐ +
๐โ๐=1
โ๐๐๐, โ๐ โฅ 0, โ๐ โ R, โ๐ > 0 for ๐ โ ๐ผ
}
โ the semigroup generated by ๐1, ๐2, . . . , ๐๐ at ๐ as
๐๐ผZ (๐1, ๐2, . . . , ๐๐; ๐) =
{๐ฅ โ R๐ : ๐ฅ = ๐ +
๐โ๐=1
โ๐๐๐, โ๐ โฅ 0, โ๐ โ Z, โ๐ > 0 for ๐ โ ๐ผ
}
โ given a lattice โ, the saturated semigroup generated by ๐1, ๐2, . . . , ๐๐ at ๐ as
๐๐ผR,โ (๐1, ๐2, . . . , ๐๐; ๐) = ๐๐ผR (๐1, ๐2, . . . , ๐๐; ๐) โฉ โ
๐R ((1, 1), (2, 0)) ๐Z ((1, 1), (2, 0)) ๐R,Z๐ ((1, 1), (2, 0))
๐R ((1, 1), (2, 0); (1, 3)) ๐Z ((1, 1), (2, 0); (1, 3)) ๐R,Z๐ ((1, 1), (2, 0); 1, 3))
1.1. POLYHEDRA, CONES AND SEMIGROUPS 29
๐{1}R ((1, 1), (2, 0); (1, 3)) ๐{1}Z ((1, 1), (2, 0); (1, 3)) ๐{1}R,Z๐ ((1, 1), (2, 0); (1, 3))
Illustration of the definitions.
๏ฟฝ
An important tool in polyhedral geometry is triangulation. A polyhedral objectwith complicated geometry can be decomposed into simpler ones. The building block isthe simplex. Since we are interested mostly in cones, we present the definition for thetriangulation of a cone.
Definition 1.14 (Triangulation of a cone)We define a triangulation of a real cone ๐ถ = ๐R (๐1, ๐2, . . . , ๐๐) as a finite collection ofsimplicial cones ฮ = {๐ถ1, ๐ถ2, . . . , ๐ถ๐ก} such that:
โโ๐ถ๐ = ๐ถ,
โ If ๐ถ โฒ โ ฮ then every face of ๐ถ โฒ is in ฮ,
โ ๐ถ๐ โฉ ๐ถ๐ is a common face of ๐ถ๐ and ๐ถ๐ .
๏ฟฝ
The following proposition says that we can triangulate a cone without introducing newrays.
Proposition 2 (see [17]). A pointed convex polyhedral cone ๐ถ admits a triangulation ฮwhose 1-dimensional cones are the extreme rays of ๐ถ. ๏ฟฝ
A concept that we will use heavily in later chapters is that of the vertex cone.
Definition 1.15 (Tangent and Feasible Cone)Let ๐ โ R๐ be a polyhedron and ๐ฃ a vertex of ๐ , the tangent cone ๐ฆ๐ฃ and the feasiblecone โฑ๐ฃ of ๐ at ๐ฃ are defined by
โฑ๐ฃ := {๐ข | โ๐ฟ > 0 : ๐ฃ + ๐ฟ๐ข โ ๐} ,๐ฆ๐ฃ := ๐ฃ + โฑ๐ฃ.
๏ฟฝ
We will call ๐ฆ๐ฃ a vertex cone.
30 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
1.2 Formal Power Series and Generating Functions
1.2.1 Formal Power Series
Starting from the univariate polynomial ring, we will climb up an hierarchy of algebraicstructures that are related to polyhedral geometry and partition analysis. For a moredetailed exposition see [34, 37] and references therein.
Let K be a field, that we can assume is the field of complex numbers. As usual, wedenote by K[๐ง] the polynomial ring in the variable ๐ง. We will generalize polynomials infour ways, namely by considering fractions of polynomials, allowing an infinite numberof terms, allowing negative exponents and considering more than one variables.
The polynomial ring K[๐ง] contains no zero-divisors, thus it is an integral domain.Since K[๐ง] is an integral domain, we can define its fraction field, i.e., the field of univariaterational functions, denoted by K(๐ง). A polynomial can be considered as a formal sumโ
๐โ๐ผ๐๐๐ง
๐ for some finite index set ๐ผ โ N and ๐๐ โ K.
If we allow as index sets infinite subsets of the natural numbers, then we obtain formalpowerseries
๐(๐ง) =
โโ๐=0
๐๐๐ง๐ for ๐๐ โ K.
Another way to see formal powerseries is as sequences. Given an infinite sequence ๐ด =[๐1, ๐2, . . .] with elements from K and a formal variable ๐ง, then ๐(๐ง) represents ๐ด. Theusual definition of formal powerseries is as sequences. Define the following two operationsof addition and (Cauchy) multiplication for two formal powerseries ๐(๐ง) and ๐(๐ง):
๐(๐ง) + ๐(๐ง) =
โโ๐=0
(๐๐ + ๐๐) ๐ง๐,
๐(๐ง)๐(๐ง) =โโ๐=0
โโ ๐โ๐=0
๐๐๐๐โ๐
โโ ๐ง๐.
The ring of formal powerseries is the set
KJ๐งK =
{ โโ๐=0
๐๐๐ง๐
๐๐ โ K
}
equipped with the above defined addition and multiplication. Note that the ring ofpolynomials is a subring of the ring of formal powerseries. If we allow for polynomialscontaining negative exponents (but only finitely many terms), then we obtain the ring ofLaurent polynomials. We denote this ring by K
[๐ง, ๐งโ1
]or K
[๐งยฑ1]. Next, we construct
the polynomial ring K[๐ฅ, ๐ฆ] as (K[๐ฆ]) [๐ฅ], i.e., the ring of polynomials in ๐ฅ with coefficientsin the ring K[๐ฆ] . We define recursively the polynomial ring in ๐ variables, denoted by
1.2. FORMAL POWER SERIES AND GENERATING FUNCTIONS 31
K[๐ง1, ๐ง2, . . . , ๐ง๐]. The fraction field of K[๐ง1, ๐ง2, . . . , ๐ง๐], which is the multivariate versionof the rational function field, is denoted by K(๐ง1, ๐ง2, . . . , ๐ง๐). As in the univariate case,allowing for negative exponents, we obtain the multivariate Laurent polynomial ring,denoted by K[๐งยฑ1
1 , ๐งยฑ12 , . . . , ๐งยฑ1
๐ ]. Like we did with polynomials, we can construct thering KJ๐ฅ, ๐ฆK of formal powerseries in ๐ฅ and ๐ฆ as the ring of formal powerseries in ๐ฅwith coefficients in KJ๐ฆK. A ring that we will use extensively in the next chapters isthat of multivariate formal powerseries in ๐ variables, defined recursively and denotedby KJ๐ง1, ๐ง2, . . . , ๐ง๐K.
An important property of the ring of formal powerseries is given by the following
Lemma 3 (see [27]). If ๐ท is an integral domain then ๐ทJ๐งK is an integral domain aswell. ๏ฟฝ
Now, let us define the field of univariate Laurent series, which can be thought ofeither as the fraction field of the ring of univariate formal powerseries or as Laurentpolynomials with finitely many terms of negative exponent but possibly infinitely manyterms of positive exponent. More formally
Definition 1.16 (Univariate formal Laurent series)The set of formal expressions
KL๐งM =
{โ๐โZ
๐๐๐ง๐
๐๐ โ K, ๐๐ = 0 for all but finitely many negative values of ๐
}
equipped with the usual addition and (Cauchy) multiplication is the field of univariateformal Laurent series. ๏ฟฝ
The multivariate equivalent is much harder to deal with. In particular, there aremany candidates as appropriate generalizations. We will explore some of them.
In [37], Xin presents two generalizations of formal Laurent series. The first and morestraightforward is that of iterated Laurent series.
Definition 1.17 (Iterated Laurent series, Section 2.1 in [37])The field of iterated Laurent series in ๐ variables is defined recursively as
Kโจโจ๐ง1, ๐ง2, . . . , ๐ง๐โฉโฉ = Kโจโจ๐ง1, ๐ง2, . . . , ๐ง๐โ1โฉโฉL๐ง๐M
where Kโจโจ๐ง1โฉโฉ = L๐ง1M. ๏ฟฝ
Next he presents Malcev-Neuman series, which allow the most general setting amongthe ones presented here.
Definition 1.18 (Malcev-Neuman series, Theorem 3-1.6 in [37])Let ๐บ be a totally ordered monoid and ๐ a commutative ring with unit. A formal series๐ on ๐บ has the form
๐ =โ๐โ๐บ
๐๐๐
32 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
where ๐๐ โ ๐ and ๐ is regarded as a symbol. The support of ๐ is defined to be theset {๐ โ ๐บ | ๐๐ = 0}. A Malcev-Neumann series is a formal series on ๐บ that has a well-ordered support. We define ๐ ๐ค[๐บ] to be the set of all such MN-series. Then ๐ ๐ค[๐บ] is aring. ๏ฟฝ
In [34], Aparicio-Monforte and Kauers introduce three algebraic structures relatedto multivariate formal Laurent series. They use polyhedral cones and the notion of acompatible additive ordering for their definitions.
Definition 1.19 (Compatible additive ordering [34])Given a pointed simplicial cone ๐ถ โ R๐ with apex at the origin, a total order โค on Z๐ iscalled additive if for all ๐, ๐, ๐ โ Z๐ we have ๐ โค ๐โ ๐+๐ โค ๐+๐. It is called compatiblewith the cone ๐ถ if 0 โค ๐ for all ๐ โ ๐ถ. ๏ฟฝ
First they define the ring of multivariate formal Laurent series supported in a cone.
Definition 1.20 (Multivariate Laurent series from cones [34])Given a pointed cone ๐ถ โ R๐, the set
K๐ถ J๐ง1, ๐ง2, . . . , ๐ง๐K =
{โ๐โ๐ถ
๐๐๐ง๐
๐๐ โ K
}
equipped with the usual addition and Cauchy multiplication forms a ring . ๏ฟฝ
A larger ring is obtained if we fix an ordering and consider the collection ๐ of all conesthat are compatible with this ordering โค. Then
Kโค J๐ง1, ๐ง2, . . . , ๐ง๐K =โ๐ถโ๐
K๐ถ J๐ง1, ๐ง2, . . . , ๐ง๐K
is a ring.
Finally, Aparicio-Monforte and Kauers construct a field of multivariate formal Lau-rent series by taking the union of translates of Kโค J๐ง1, ๐ง2, . . . , ๐ง๐K to all lattice points.
KโคL๐ง1, ๐ง2, . . . , ๐ง๐M =โ๐โZ๐
๐ง๐Kโค J๐ง1, ๐ง2, . . . , ๐ง๐K .
The lattice points of a pointed cone form a well-ordered set. Moreover, given theadditive compatible ordering, required by the construction of Kโค J๐ง1, ๐ง2, . . . , ๐ง๐K, the sup-port of each object in Kโค J๐ง1, ๐ง2, . . . , ๐ง๐K is well-ordered (and 0 is the minimum). Finally,in the construction of KโคL๐ง1, ๐ง2, . . . , ๐ง๐M the supports appearing in KโคJ๐ง1, ๐ง2, . . . , ๐ง๐K maybe translated, but this does not affect the fact that they are well-ordered. This meansthat these three constructions are examples of Malcev-Neuman series, as noted in [34].
Now we will put into the picture of the algebraic structures the sets of power seriesand rational functions occurring in the geometric context. The generating function forthe lattice points of a rational simplicial polyhedral cone is a rational function, see nextsection.
1.2. FORMAL POWER SERIES AND GENERATING FUNCTIONS 33
Definition 1.21 (Polyhedral Laurent Series [16])The space ๐๐ฟ of polyhedral Laurent series is the K[๐งยฑ1
1 , ๐งยฑ12 , . . . , ๐งยฑ1
๐ ]-submodule ofK[[๐งยฑ1
1 , ๐งยฑ12 , . . . , ๐งยฑ1
๐ ]] generated by the set of formal seriesโงโชโจโชโฉโ
๐ โ(๐ถโฉZ๐)
๐ง๐
๐ถ is a simplicial rational cone
โซโชโฌโชโญ .
๏ฟฝ
Since any cone can be triangulated by using only simplicial cones, we have that PLcontains the formal power series expressions of the generating functions of all rationalcones. Due to the vertex cone decomposition of polyhedra (see Section 1.3.2), ๐๐ฟ alsocontains the generating functions of all polyhedra.
The following diagram presents the relations of the structures discussed above.
34 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
[๐งยฑ1 , ๐งยฑ2 , . . . , ๐งยฑ๐ ]
Malcev-Neuman
Kโจโจ๐ง1, ๐ง2, . . . , ๐ง๐โฉโฉ KโฅL๐ง1, ๐ง2, . . . , ๐ง๐M
K๐ถJ๐ง1, ๐ง2, . . . , ๐ง๐KK (๐ง1, ๐ง2, . . . , ๐ง๐)
KJ๐ง1, ๐ง2, . . . , ๐ง๐K
KL๐งM
K[๐บ]
๐๐ฟ
K[๐งยฑ11 , ๐งยฑ1
2 , . . . , ๐งยฑ1๐ ]
Generatingfunctions of
cones
K(๐ง)
K[๐ง1, ๐ง2, . . . , ๐ง๐]
KJ๐งK
K[๐งยฑ1]
K[๐ง]
Relations of algebraic structures related to polynomials, formal power series andrational functions.
1.2. FORMAL POWER SERIES AND GENERATING FUNCTIONS 35
1.2.2 Generating Functions
One of the most important tools in combinatorics and number theory, when dealingwith infinite sequences, is that of generating functions. The hope is that a generatingfunction, although encoding full information of an infinite object, has a short (or at leastfinite) representation. We will restrict the definitions presented here in a way that coversour use of generating functions without introducing generality that reduces clarity. Fora detailed introduction see [27, 36].
The generating function is a tool for representing infinite sequences in a handy way.A sequence is a map from N to a set of values. We will assume that our values comefrom a field K, but we note that for the most part of the thesis, the only possible valuesare 0 and ยฑ1. The usual notation for a sequence is ๐ด = [๐0, ๐1, ๐2, . . .] = (๐๐)๐โN. Wedefine the generating function of ๐ด as the formal sum
ฮฆ๐ด =โโ๐=0
๐๐๐ง๐
This is a formal powerseries in KJ๐งK.If we want to compute the generating function of a set ๐, subset of the natural
numbers, we can use the above definition by considering the sequence ๐ด = [๐0, ๐1, ๐2, . . .],where ๐๐ = 1 if ๐ โ ๐ and ๐๐ = 0 otherwise. More formally, we consider the indicatorfunction of the set ๐, defined as
[๐](๐ฅ) =
{1 : ๐ฅ โ ๐,0 : ๐ฅ /โ ๐.
and then ๐ด = ([๐](๐))๐โN. We will use the notation ฮฆ๐ for the generating function ofthe set ๐, where ฮฆ๐ = ฮฆ๐ด for ๐ด = ([๐](๐))๐โN. Naturally, the first example is the naturalnumbers:
ฮฆN(๐ง) =โ๐โN
[N](๐)๐ง๐ =โ๐โN
๐ง๐ =1
(1โ ๐ง).
A slight variation is the sequence of even natural numbers. Then we have
ฮฆ2N(๐ง) =โ๐โN
[2N](๐)๐ง๐ =โ๐โN
๐ง2๐ =1
(1โ ๐ง2).
Although these two examples look too simple, it is their multivariate versions that coverthe majority of the cases we are interested in.
With the above definition we can use generating functions to deal with subsets ofthe natural numbers, but this is not sufficient for our purposes. We need to be able todescribe sets containing negative integers. Thus, we extend the definition for subsets ofZ. Given a set ๐ โ Z, we define
ฮฆ๐ =โ๐โ๐
๐ง๐.
36 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
This is not a formal powerseries anymore and depending on the set ๐ it may not be aformal Laurent series either.
Although univariate generating functions are very useful for counting, for our pur-poses multivariate ones are essential. The generalization is straightforward.
Definition 1.22 (Generating Function)Given a set ๐ โ Z๐, we define the generating function of ๐ as the formal sum
ฮฆ๐ =โ๐โZ๐
[๐](๐)๐ง๐.
๏ฟฝ
The formal sum representation of a set is no more than syntactic sugar. The represen-tation of a set of numbers or of its formal sum representation have the same (potentially)infinite number of terms. As already stated, the expectation is to have a more condensedrepresentation. A glimpse on that was given in the example of the generating functionof the natural numbers, through the use of the geometric series expansion formula. Weproceed now more formally and setting the necessary notation, defining what a rationalgenerating function is.
Definition 1.23 (Rational Generating Function)Let ๐ โ Z๐. If there exists a rational function in K(๐ง1, ๐ง2, . . . , ๐ง๐), which has a seriesexpansion equal to ฮฆ๐ , then we denote that rational function by ๐๐ (๐ง1, ๐ง2, . . . , ๐ง๐) andcall it a rational generating function of ๐. ๏ฟฝ
Here one should note that we make no assumptions on the form of the rationalfunction (e.g. reduced, the denominator has a specific form etc.). This means that aformal powerseries (or formal Laurent series) may have more than one rational functionforms. In the other way, the same rational function can have more than one seriesexpansions. In other words, although the formal sum generating function of a set is a welldefined object, the rational generating function of a set is subject to more parameters.It is not well defined (yet), but extremely useful. We will heavily use rational generatingfunctions and in Section 1.3 we will explain why we are on safe ground.
1.2.3 Generating Functions for Semigroups
In this thesis we deal with generating functions related to polyhedra. For this, it is suffi-cient to compute with generating functions of cones. There are two types of semigroupsthat are of interest for us, the discrete semigroup generated by a set of integer vectorsand the corresponding saturated semigroup. We first compute the generating functionof the semigroup ๐ = ๐Z ((1, 3), (1, 0)).
1.2. FORMAL POWER SERIES AND GENERATING FUNCTIONS 37
๐Z (N) ((1, 3), (1, 0)).The generating function as a formal power series is
ฮฆ๐(z) =โ
iโ{๐(1,3)+โ(1,0)|๐,โโN}
zi =โ๐,โโN
z๐(1,3)+โ(1,0) =
(โ๐โN
z๐(1,3)
)(โโโN
zโ(1,0)
).
Thus, by the geometric series expansion formula we have
๐๐(z) =
(1
1โ z(1,3)
)(1
1โ z(1,0)
)=
1(1โ ๐ง1๐ง32
)(1โ ๐ง1)
.
In general it is easy to see that if a1,a2, . . . ,an are linearly independent, then for thesemigroup ๐ = ๐Z (a1,a2, . . . ,an) we have
๐๐(z) =1
(1โ za1) (1โ za2) ยท ยท ยท (1โ zan).
This is true because in the expansion of 1(1โza1 )(1โza2 )ยทยทยท(1โzan ) we get as exponents all
non-negative integer combinations of the exponents in the denominator.Observe that if we translate the semigroup to a lattice point, then the structure does
not change at all.
๐Z (N) ((1, 3), (1, 0); (1, 1)).
38 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
There is a bijection between ๐Z ((1, 3), (1, 0)) and ๐Z ((1, 3), (1, 0; (1, 1)), given by ๐(s) =s+ (1, 1). Translating this to generating functions we have that
ฮฆ๐Z((1,3),(1,0);(1,1))(z) =
โiโ{๐(1,3)+โ(1,0)+(1,1)|๐,โโN}
zi
=โ๐,โโN
z๐(1,3)+โ(1,0)+(1,1)
= z(1,1)
(โ๐โN
z๐(1,3)
)(โโโN
zโ(1,0)
).
Now it is clear that the generating function for ๐Z ((1, 3), (1, 0; (1, 1)) is rational , namely
๐๐Z((1,3),(1,0);(1,1))(z) = z(1,1)
(1
1โ z(1,3)
)(1
1โ z(1,0)
)=
๐ง1๐ง2(1โ ๐ง1๐ง32
)(1โ ๐ง1)
.
In general for a semigroup ๐ = ๐Z (a1,a2, . . . ,an;q) we have
๐๐(z) =zq
(1โ za1) (1โ za2) ยท ยท ยท (1โ zan).
Saturated Semigroup
Letโs consider the saturated semigroup ๐ = ๐R,Z๐ ((1, 3), (1, 0)), and compute its gener-
ating function as a formal powerseries ฮฆ๐ =โ๐ โ๐
z๐ .
๐R,Z๐ ((1, 3), (1, 0)).
We observe in the following figure that ๐ can be partitioned in three sets. The bluepoints are reachable starting from the origin and using only the cone generators. Thered points are reachable by using only the cone generators if we start from the point(1, 1). And similarly for the green ones if we start from (1, 2).
1.2. FORMAL POWER SERIES AND GENERATING FUNCTIONS 39
The saturated semigroup separated in three subsets.
This provides a direct sum decomposition
๐R,Z๐ ((1, 3), (1, 0)) = ๐Z ((1, 3), (1, 0))โ ๐Z ((1, 3), (1, 0); (1, 1))โ ๐Z ((1, 3), (1, 0); (1, 2)) .
In general we have the following relation:
๐R,Z๐ (a1,a2, . . . ,an) = โqโฮ Z๐(๐R(a1,a2,...,an))๐Z (a1,a2, . . . ,an;q) .
If the apex ๐ of the cone of the saturated semigroup is not the origin, we have to translateevery lattice point by ๐, thus multiply the generating function by zq.From the relation above and the form of the rational generating function of a discretesemigroup ๐Z (a1,a2, . . . ,an;q) we immediately deduce the lemma:
Lemma 4. For a saturated semigroup ๐ = ๐R,Z๐ (a1,a2, . . . ,an; ๐) we have
๐๐(z) =๐ง๐โ
๐ผโฮ Z๐(๐R(a1,a2,...,an)) z
๐ผ
(1โ za1) (1โ za2) ยท ยท ยท (1โ zan).
๏ฟฝ
Generating Functions of Open Cones
Given a simplicial cone
๐๐ผR (a1,a2, . . . ,ak;q) =
{๐ฅ โ R๐ : ๐ฅ = ๐ +
๐โ๐=1
โ๐a๐, โ๐ โ N, โ๐ > 0 for ๐ โ ๐ผ
}we can apply the inclusion exclusion principle to get an expression of the open cone asa signed sum of closed cones
๐๐ผR (a1,a2, . . . ,ak;q) =โ๐โ๐ผ
(โ1)|๐ |๐R((ai)๐โ๐ผโ๐ ;q
).
40 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
Example 1. Given the half-open cone ๐{1,2}R ((1, 0, 0), (0, 1, 0), (0, 0, 1)) we have
๐{1,2}R ((1, 0, 0), (0, 1, 0), (0, 0, 1)) = ๐R ((1, 0, 0), (0, 1, 0), (0, 0, 1))
โ (๐R ((1, 0, 0), (0, 0, 1)) + ๐R ((0, 1, 0), (0, 0, 1)))
+ ๐R ((0, 0, 1)) .
๐ฅ
๐ฆ
๐1
๐2
๐3
{1}
โจ(1, 0, 0), (0, 0, 1)โฉ
{2}
{1, 2}
Inclusion-exclusion for ๐{1,2}R ((1, 0, 0), (0, 1, 0), (0, 0, 1); 0).
๏ฟฝ
1.3. FORMAL SERIES, RATIONAL FUNCTIONS AND GEOMETRY 41
1.3 Formal Series, Rational Functions and Geometry
In this section we review some facts from polyhedral geometry and generating functionsof polyhedra. The main goal is to arrive to a consistent correspondence between ratio-nal generating functions for polyhedra and formal powerseries. This is done throughthe definition of expansion directions. In the end of the section we present two funda-mental theorems from polyhedral geometry for the decomposition of rational generatingfunctions and shortly review Barvinokโs algorithm.
1.3.1 Indicator and rational generating functions
We start by the definition of the space of indicator functions of rational polyhedra.
Definition 1.24 (from [11])The real vector space spanned by indicator functions [๐ ] of rational polyhedra ๐ โ R๐
is called the algebra of rational polyhedra in R๐ and is denoted by ๐ซQ(R๐). ๏ฟฝ
Theorem 3.3 in [11] is essential for the connection of geometry with rational functions.
Theorem 1.2 (VIII.3.3 in [11])There exists a map
๐ : ๐ซQ(R๐)โ C (๐ง1, ๐ง2, . . . , ๐ง๐)
such that
โ ๐ is a linear transformation (valuation).
โ If ๐ โ R๐ is a rational polyhedron without lines, then ๐ [๐ ] = ๐๐ (z) such that๐๐โฉZ๐(z) = ฮฆ
๐โฉZ๐(z), provided that ฮฆ๐โฉZ๐(z) converges absolutely.
โ For a function ๐ โ ๐ซQ(R๐)and q โ Z๐, let โ(z) = ๐(zโ q) be a shift of ๐. Then
๐โ = zq๐๐.
โ If ๐ โ R๐ is a rational polyhedron containing a line, then ๐ [๐ ] โก 0.
๏ฟฝ
Let us see how this map works in an example.
Example 2. Let ๐ = 1. We consider the polyhedra:
โ ๐+ = R+ (including the origin)
โ ๐โ = Rโ (including the origin)
โ ๐ = R (the real line)
โ ๐0 = {0}
The associated generating functions (converging in appropriate regions) are:
42 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
โ ฮฆ๐+โฉZ(๐ง) =โ
๐โ๐+โฉZ๐ง๐ =
โโ๐=0
๐ง๐ =1
1โ ๐ง= ๐๐+โฉZ(๐ง)
โ ฮฆ๐โโฉZ(๐ง) =โ
๐โ๐โโฉZ๐ง๐ =
0โ๐=โโ
๐ง๐ =1
1โ ๐งโ1= ๐๐โโฉZ(๐ง)
โ ฮฆ๐0โฉZ(๐ง) =โ
๐โ๐0โฉZ๐ง๐ = ๐ฅ0 = 1 = ๐๐0โฉZ(๐ง)
From Theorem 1.2 we have that:
โ ๐ [๐+] =1
1โ๐ง
โ ๐ [๐โ] =1
1โ๐งโ1
โ ๐ [๐0] = 1
By inclusion-exclusion we have
[๐ ] = [๐+] + [๐โ]โ [๐0]
Thus one expects that
0 = ๐ [๐ ] = ๐ [๐+] + ๐ [๐โ]โ ๐ [๐0] =1
1โ ๐ง+
1
1โ ๐งโ1โ 1
which is indeed the case. In other words
๐๐โฉZ(๐ง) = ๐๐+โฉZ(๐ง) + ๐๐โโฉZ(๐ง)โ ๐๐0โฉZ(๐ง).
๏ฟฝ
Example 3. Compute the integer points of ๐ = [๐, ๐] โ R for ๐ < ๐. Let ๐1 = [โโ, ๐]and ๐2 = [๐,โ]. Then
[๐ ] = [๐1] + [๐2]โ [R]
and
โ ฮฆ๐1โฉZ(๐ง) =โโ
๐=๐ ๐ง๐ = ๐ง๐
1โ๐ง โ ๐ [๐1] =๐ง๐
1โ๐ง = ๐๐1โฉZ(๐ง)
โ ฮฆ๐2โฉZ(๐ง) =โ๐
๐=โโ ๐ง๐ = ๐ง๐
1โ๐งโ1 โ ๐ [๐2] =๐ง๐
1โ๐งโ1 = ๐๐2โฉZ(๐ง)
โ ๐ [R] = 0 = ๐RโฉZ(๐ง)
Thus
๐๐โฉZ =๐ง๐
1โ ๐ง+
๐ง๐
1โ ๐งโ1=
๐ง๐ โ ๐ง๐+1
1โ ๐ง
1.3. FORMAL SERIES, RATIONAL FUNCTIONS AND GEOMETRY 43
This is expected because๐ง๐
1โ ๐งis the generating function of the ray starting at ๐, while
๐ง๐+1
1โ ๐งis the generating function of the ray starting at ๐+1. Subtracting the second from
the first we obtain the segment ๐ .If we evaluate ๐๐โฉZ(๐ง) at ๐ง = 1 (using de lโHospitalโs rule) we get ๐ โ ๐ + 1, the
number of lattice points in ๐ . ๏ฟฝ
Theorem 1.2 provides a correspondence between different representations of gener-ating functions of polyhedra.
Given an indicator function ๐ : R๐ โ {0, 1}, ๐ โ ๐ซQ(R๐), we consider its restriction
๐โฒ : Z๐ โ {0, 1}. We denote by ๐ซQ(Z๐)the set of restricted indicator functions.
๐ซ๐R ๐ซQ
(R๐)
๐ซ๐Z ๐ซQ
(Z๐)
KโฅJ๐งK K(๐ง)
๐1
๐2
๐3
๐4
๐5
๐
๐6
๐7
๐8
Figure 1.1: Polyhedra, indicator functions, Laurent and rational generating functions.
The arrows in Figure 1.1 have the following meaning:
๐1 Bijection between polyhedra in R๐ and their indicator functions.
๐2 Map a polyhedron ๐ โ R๐ to ๐ โฉ Z๐ .
๐3 Bijection between sets of lattice points of polyhedra and their Laurent series gen-erating functions.
๐4 Bijection between lattice points in polyhedra and their restricted indicator func-tions.
๐5 Bijection between restricted indicator function of polyhedra and their Laurentseries generating functions.
๐6 Map the restricted indicator function of a polyhedron to a rational function.
๐7 Restriction.
๐8 Map a Laurent series with support in a polyhedron to a rational function.
44 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
In order to be able to map multivariate Laurent series to rational functions we restrictto series in KโฅJ๐งK, i.e., we assume that the series have support in a pointed cone [34].The most important maps are ๐8 and ๐6, but unfortunately they are not bijective. Wewill first see why and then fix this bijection.
When trying to establish the connection between formal powerseries, rational func-tions and polyhedral geometry we encounter problems due to the one-to-many relationbetween rational functions and their Laurent expansions.
Example 4 ([23]). Consider the finite set ๐ = {0, 1, 2, . . . , ๐} โ N. Its rational gener-ating function is the truncated geometric series
๐๐(๐ง) =1โ ๐ง๐+1
1โ ๐ง
Now we can rewrite this expression as
๐๐(๐ง) =1
(1โ ๐ง)โ ๐ง๐+1
(1โ ๐ง)
=1
(1โ ๐ง)+
๐ง๐
(1โ ๐งโ1)
and we can expand each summand using the geometric series expansion formula.
๐๐(๐ง) =1
(1โ ๐ง)+
๐ง๐
(1โ ๐งโ1)
=(๐ง0 + ๐ง1 + ๐ง2 + . . .
)+(๐ง๐ + ๐ง๐โ1 + ๐ง๐โ2 + . . .
)= 2ฮฆ๐(๐ง) +
โโ๐=๐+1
๐ง๐ +โ1โ
๐=โโ๐ง๐.
If we interpret each of the summands geometrically we end up with the required segmentcounted twice and the rest of the line counted once, which is not what we expected. Thisis due to the fact that the power series expansions around 0 andโ of the two summands,viewed as analytic functions, have disjoint regions of convergence. In particular we have:
1
(1โ ๐ง)=
{๐ง0 + ๐ง1 + ๐ง2 + ยท ยท ยท for|๐ง| < 1,โ๐งโ1 โ ๐งโ2 โ ๐งโ3 ยท ยท ยท for|๐ง| > 1
and
๐ง๐
1โ ๐งโ1=
{โ๐ง๐+1 โ ๐ง๐+2 โ ๐ง๐+3 ยท ยท ยท for|๐ง| < 1,๐ง๐ + ๐ง๐โ1 + ๐ง๐โ2 ยท ยท ยท for|๐ง| > 1.
Thus, only the choices where the region of convergence coincide seem natural:
1.3. FORMAL SERIES, RATIONAL FUNCTIONS AND GEOMETRY 45
๐๐(๐ง) =1
(1โ ๐ง)+
๐ง๐
(1โ ๐งโ1)
=(๐ง0 + ๐ง1 + ๐ง2 + ยท ยท ยท
)+(โ๐ง๐+1 โ ๐ง๐+2 โ ๐ง๐+3 ยท ยท ยท
)= ฮฆ๐(๐ง)
or
๐๐(๐ง) =1
(1โ ๐ง)+
๐ง๐
(1โ ๐งโ1)
=(โ๐งโ1 โ ๐งโ2 โ ๐งโ3 ยท ยท ยท
)+(๐ง๐ + ๐ง๐โ1 + ๐ง๐โ2 ยท ยท ยท
)= ฮฆ๐(๐ง).
๏ฟฝ
Moving from the rational generating function of a set to its geometric representation,we are fixing an expansion direction. This is implicitly done through choosing theLaurent expansion we will interpret geometrically. In Example 4 essentially we pick adirection (either towards positive infinity of negative infinity) and consistently take seriesexpansions with respect to this direction. Even if each of the summands have a perfectlymeaningful expansion with respect to any direction, when we want to make sense out oftheir sum (or simultaneously consider their geometry) we need a consistent choice. Thefollowing example shows how this is important for our purposes.
Example 5. Consider the rational function ๐(๐ง) = ๐ง3
(1โ๐ง) . One can take either theโforwardโ or the โbackwardโ expansion of ๐ as Laurent power series corresponding toexpansions around 0 or โ.
๐ง3
(1โ ๐ง)=
{๐ง3 + ๐ง4 + ๐ง5 + ๐ง6 + ยท ยท ยท = ๐น1
โ๐ง2 โ ๐ง1 โ ๐ง0 โ ๐งโ1 โ ยท ยท ยท = ๐น2
If we denote by ๐น โฒ๐ the power series having only the terms of ๐น๐ that have non-negative
exponents then๐น โฒ1 = ๐ง3 + ๐ง4 + ๐ง5 + ๐ง6 + ยท ยท ยท = ฮฆ[3,โ](๐ง)
and๐น โฒ2 = โ๐ง2 โ ๐ง1 โ ๐ง0 = โฮฆ[0,2](๐ง)
Observe that the two resulting rational functions ๐[3,โ](๐ง) and โ๐[0,2](๐ง) are different,one is a polynomial and the other is not, although ๐น1 and ๐น2 are expansions of the samerational function. ๏ฟฝ
The operation of keeping only the terms with non-negative exponents is a specialkind of taking intersections in terms of geometry. Moreover, it is an essential part of thecomputation of ฮฉโฅ in a geometric way (see Section 3.1).
46 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
Now we will define the notion of forward expansion direction more formally. Forbi โ R๐, let ๐ต = {b1,b2, . . . ,bn} be an ordered basis of R๐ and
๐ป๐ต =
{โ๐
โ๐bi : โ๐ โ R, if ๐ โ [๐] is minimal such that โ๐ = 0 then โ๐ > 0
}โช {0}.
We need the notion of the recession cone in order to define directions.
Definition 1.25 (Recession Cone)Given a polyhedron ๐ , its recession cone is defined as
rec(๐ ) = {๐ฆ โ R๐ : ๐ฅ+ ๐ก๐ฆ โ ๐ for all ๐ฅ โ ๐ and for all ๐ก โฅ 0}
๏ฟฝ
In order for a Laurent series with support ๐ to be well defined we require that thereexists a polyhedron ๐ such that ๐ โ ๐ and rec(๐ ) โ ๐ป๐ต, for some ordered basis ๐ต. Wesay that ๐ป๐ต defines the โforward directionsโ, in the following sense.
Definition 1.26 (Forward Direction)For any ordered basis ๐ต of R๐ we call ๐ฃ โ R๐ forward if and only if ๐ฃ โ ๐ป๐ต. ๏ฟฝ
Note that for every ๐ฃ โ Z๐ โ {0} either ๐ฃ or โ๐ฃ is forward. Now we can fix theproblem we had in the diagram about correspondence of generating functions.
Lemma 5. Fix an ordered basis. If Laurent series expansions are taken, consistently,only with respect to forward directions, then there is a bijection between Laurent seriesgenerating functions of lattice points of polyhedra and the respective rational generatingfunctions. ๏ฟฝ
From now on, we will use series expansions using only forward directions.
Example 6. Let ๐1 = [๐,โ] and ๐2 = [โโ, ๐]. We want to compute ฮฆ๐1+๐2(๐ง) and
๐๐1+๐2(๐ง).
The forward directions are given by the basis ๐ต = {1} of R and thus ๐ป๐ต containsall vectors that have positive first coordinate, i.e., ๐ฃ โ R with ๐ฃ > 0. In other words, weaccept any Laurent series that are (possibly) infinite towards โ, but not towards โโ.
We have
ฮฆ๐1โฉZ(๐ง) =
โโ๐=๐
๐ง๐ =๐ง๐
1โ ๐งโ ๐ [๐1] =
๐ง๐
1โ ๐ง= ๐๐1โฉZ(๐ง)
but for the second cone, the expansion should be consistent, i.e., towards โ. Thus wewill use ๐3 = [๐+ 1,โ] and the fact that ๐2 = Rโ ๐3. We know that [๐2] = [R]โ [๐3]since ๐2 โฉ ๐3 = โ . From Theorem 1.2 we know that ๐ [R] = 0, thus [๐2] = โ[๐3]. Thisimplies that in the computation of the rational generating function we can ignore R.
ฮฆ๐2โฉZ(๐ง) = ฮฆRโฉZ(๐ง)โ ฮฆ๐3โฉZ(๐ง) =
โโ๐=โโ
๐ง๐ โโโ
๐=๐+1
๐ง๐
1.3. FORMAL SERIES, RATIONAL FUNCTIONS AND GEOMETRY 47
โผ โโโ
๐=๐+1
๐ง๐ = โ ๐ง๐+1
1โ ๐งโ ๐ [๐2] = โ
๐ง๐+1
1โ ๐ง= ๐๐2โฉZ(๐ง).
โผ means that the two Laurent series correspond to polyhedra with the same rationalfunction. Geometrically this means to โflipโ the ray from pointing to โโ to pointing toโ.
Thus
๐๐โฉZ(๐ง) =๐ง๐
1โ ๐งโ ๐ง๐+1
1โ ๐ง=
๐ง๐ โ ๐ง๐+1
1โ ๐ง.
๏ฟฝ
The reason why this phenomenon is not a problem for the classical ฮฉโฅ algorithms (seeSection 3.1) is that by the construction of the crude generating function, we make a choiceof expansion directions for all ๐ variables. In particular, when converting the multisumexpression into a rational function, we chose to consider only โforwardโ expansions forall variables (where forward is defined with respect to the standard basis).
1.3.2 Vertex cone decompositions
We present two theorems that provide formulas for writing the rational generating func-tion of a polyhedron as a sum of rational generating functions of vertex cones. The maindifference between the two formulas is that in the Lawrence-Varchenko formula all conesinvolved are considered as forward expansions, while in Brionโs formula this is not true.
As a matter of fact, Brionโs formula holds true in the indicator functions level modingout lines, which is equivalent to changing the direction of a ray. When reading theformula in the rational function level though, this is not visible since the generatingfunctions of cones obtained by flipping directions are different as Laurent series but thesame as rational functions.
Theorem 1.3 (Brion, 4.5 in [11])Let ๐ โ R๐ be a rational polyhedron and ๐ฆ๐ฃ be the tangent cone of ๐ at vertex ๐ฃ. Then
[๐ ] =โ
๐ฃ vertex of ๐
[๐ฆ๐ฃ] mod polyhedra containing lines
๏ฟฝ
Observe that from a general inclusion exclusion formula for indicator functions of tangentcones, if we mod out polyhedra containing lines, then only the vertex cones survive. Asa corollary, applying the map of Theorem1.2, we obtain
Proposition 3 (see [17]). If ๐ is a rational polyhedron, then
๐๐ (z) =โ
๐ฃ vertex of ๐
๐๐ฆ๐ฃ(z)
๏ฟฝ
48 CHAPTER 1. GEOMETRY, ALGEBRA AND GENERATING FUNCTIONS
A similar theorem is given by Lawrence and Varchenko (independently), see [16].Assume ๐ is a simple polytope in R๐ and fix a direction ๐ โ R๐ that is not perpendicularto any edge of ๐ . Denote by ๐ธ+
๐ฃ (๐) the set of edge directions ๐ค at the vertex ๐ฃ suchthat ๐ค ยท ๐ > 0 and ๐ธโ
๐ฃ (๐) the set of edge directions ๐ค such that ๐ค ยท ๐ < 0. We definethe vertex cones at ๐ฃ by flipping the direction of the edges that belong in ๐ธโ
๐ฃ (๐). Moreprecisely
๐พ๐๐ฃ = ๐ฃ +
โ๐คโ๐ธ+
๐ฃ (๐)
Rโฅ0๐ค +โ
๐คโ๐ธโ๐ฃ (๐)
R<0๐ค
Theorem 1.4 (Lawrence-Varchenko, [16])At the rational generating functions level we have
๐๐ (z) =โ
๐ฃ a vertex in ๐
(โ1)|๐ธโ๐ฃ (๐)|๐
๐พ๐๐ฃ(z)
๏ฟฝ
In other words, for each flip when constructing the vertex cone we multiply the rationalgenerating function of the cone by โ1. In the following figures, we see in action theapplication of a vertex cone decomposition for a pentagon. 2
-
-
-
-
-
+
1.3.3 Barvinokโs Algorithm
Barvinokโs algorithm [12, 28], if the dimension is fixed, is a polynomial-time algorithmthat computes the rational generating function of a rational polyhedron.
The main idea behind the algorithm is to compute a unimodular decomposition ofthe given polyhedron. To this end, we first triangulate in order to obtain simplicialcones. If we try to decompose a simplicial cone ๐ถ into cones that sum up to ๐ถ (possiblyusing inclusion-exclusion), in the worst case, we will need as many cones as the numberof lattice points in the fundamental parallelepiped of the cone. This means that insteadof having a potentially exponentially big numerator polynomial, we have a sum of verysimple rational functions (with numerator equal to 1), but with potentially exponentiallymany summands.
2These figures were shown to me by Felix Breuer.
1.3. FORMAL SERIES, RATIONAL FUNCTIONS AND GEOMETRY 49
Barvinok proposed the use of signed decompositions. Instead of decomposing thecone ๐ถ into unimodular cones that are contained in ๐ถ, one can find a unimodulardecomposition where cones with generators in the exterior of ๐ถ are involved. In thatcase, the generating functions of the unimodular cones involved in the sum will have asign. The advantage of such a signed unimodular decomposition is that it is computablein polynomial time (in fixed dimension), which implies that the number of unimodularcones is not exponential as well.
Algorithm 1 Summary of Barvinokโs algorithm
Require: ๐ โ R๐ is a rational polyhedron in h-representation
1: Compute the vertices of ๐ , denoted by ๐ฃ๐.2: Compute the vertex cones ๐๐ฃ๐ .3: If necessary, triangulate ๐๐ฃ๐ .4: Compute a signed unimodular decomposition (find a shortest vector).5: Compute the lattice points in the fundamental parallelepiped of each cone.6: return
โ๐=1,2...,(number of cones) ๐๐
1โ๐๐=1
(1โzb
]ij) , where ๐๐ is the sign of the ๐-th cone
in the signed decomposition and ๐๐1, ๐๐2 . . . , ๐๐๐ are the generators of the ๐-th cone.
Chapter 2
Linear Diophantine Systems
โHere lies Diophantus,โ the wonderbehold. Through art algebraic, thestone tells how old: โGod gave him hisboyhood one-sixth of his life, Onetwelfth more as youth while whiskersgrew rife; And then yet one-seventhere marriage begun; In five yearsthere came a bouncing new son. Alas,the dear child of master and sageAfter attaining half the measure ofhis fatherโs life chill fate took him.After consoling his fate by the scienceof numbers for four years, he endedhis life.โ
Diophantus tombstone
The main focus of this thesis is on the solution of linear Diophantine systems. Inthis chapter, we introduce some of their properties and provide a classification that isuseful when considering algorithms for solving linear Diophantine systems.
2.1 Introduction
Definition
As the name suggests we have a linear system of equations/inequalities. From linearalgebra, the standard representation of linear systems is in matrix notation. Since werespect Diophantusโ viewpoint, our matrices will always be in Z๐ร๐. In addition, thereis the restriction that the solutions are non-negative integers.
Let M be the set Z ร [๐ก1, ๐ก2, . . . , ๐ก๐], where [๐ก1, ๐ก2, . . . , ๐ก๐] denotes the multiplicativemonoid generated by ๐ก1, ๐ก2, . . . , ๐ก๐. In other words, M is the set of monomials in the
51
52 CHAPTER 2. LINEAR DIOPHANTINE SYSTEMS
variables ๐ก๐ with integer coefficients. We first give the definition of a linear Diophantinesystem.
Definition 2.1Given ๐ด โ Z๐ร๐, ๐ โ M๐ and โฆ โ {โฅ,=}, the triple (๐ด, ๐,โฆ) is called a linear Dio-phantine system. Note that the relation is considered componentwise (one inequal-ity/equation per row). Let ๐ = {๐ฅ โ N๐|๐ด๐ฅโฆ๐}. ๏ฟฝ
We denote the family of solution sets ๐ as ๐(t) in order to indicate the dependenceon the parameters. Moreover, we will denote by |๐(t)| the function on t mapping eachset in the family to its cardinality. We will refer to the family of sets ๐(t) as the solutionset of the linear Diophantine system.
Assuming that ๐ lives in (Zร [๐ก1, ๐ก2, . . . , ๐ก๐])๐ is essential for the understanding of a
certain category of problems, but for many interesting problems one can restrict tohaving constant right-hand side, i.e., ๐ โ Z๐. In the latter case, we use ๐ and |๐|, sincethe family contains only one member.
Characteristics
A linear Diophantine system has two important characteristics. These are related toits geometry and influence greatly the algorithmic treatment. A linear Diophantinesystem defines a polyhedron, whose lattice points are the solutions to the system. If thepolyhedron is bounded, i.e, it is a polytope, then we say that the linear Diophantinesystem is bounded. If the polyhedron is parametric, we say that the linear Diophantinesystem is parametric. We will clarify the notion of parametric with two examples.
The (vector partition function) problem((1 32 1
),
(๐1๐2
),โฅ)
is parametric, since the right hand side contains (honest) elements of Z ร [๐1, ๐2]. Thechoice of ๐1 and ๐2 can change considerably the geometry of the problem. The changein geometry is apparent for ๐1 = 4, ๐2 = 9 and ๐1 = 4, ๐2 = 3 in the following figures.
0 2 4 6 8 10
0
2
4
6
8
0 2 4 6 8 10
0
2
4
6
8
2.1. INTRODUCTION 53
On the other hand, the problem (of symmetric 3ร 3 magic squares with magic sum ๐ก)โโโโ 1 1 1 0 0 00 1 0 1 1 00 0 1 0 1 1
โโ ,
โโ ๐ก๐ก๐ก
โโ ,=
โโ is not a parametric problem, since the (positive integer) parameter ๐ก does not changethe geometry of the problem (every element of the right hand side is a non-constantunivariate monomial in ๐ก). In other words, we can consider the polytope ๐ , defined bythe linear Diophantine systemโโโโ 1 1 1 0 0 0
0 1 0 1 1 00 0 1 0 1 1
โโ ,
โโ 111
โโ ,=
โโ and the problem is restated as counting (or listing) lattice points in ๐ก๐ , i.e., in dilationsof ๐ . Thus ๐ก is a dilation parameter and is not changing the geometry of the problem.
It is important for any further analysis to define properly the size of a linear Dio-phantine problem. There are three input size quantities that are all relevant concerningthe size of a problem:
โ ๐ต is the number of bits needed to represent the matrix ๐ด.
โ ๐ is the number of variables.
โ ๐ is the number of relations (equations/inequalities).
In the number of relations we do not count the non-negativity constraints for thevariables. In the number of variables we do not count parameters in parametric problems.These two rules are essential in order to differentiate between problems and to defineproblem equivalence.
Counting vs Listing
Given a linear Diophantine system, one can ask two questions:
โ How many solutions are there?
โ What are the solutions?
The first is called the counting problem, while the second is the listing problem.We are interested in solving these two problems in an efficient way. The listing of thesolutions may be exponentially big in comparison to the input size ๐ต (or even infinite),thus we need a representation of the answer that encodes the solutions in an efficientway. We resort to the use of generating functions for that reason. More precisely, thesolution to the listing problem is a full (or multivariate) generating function, while forthe counting problem is a counting generating function.
54 CHAPTER 2. LINEAR DIOPHANTINE SYSTEMS
In the literature, depending on the motivation of each author, the problem specifi-cation is (sometimes silently) altered. In order to tackle the problem in an algorithmicway, we have to first resolve the specification issue. The formal definitions we use forthe two problems we are interested in are
Definition 2.2 (Listing Linear Diophantine Problem)Given a linear Diophantine system (๐ด, ๐,โฆ) โ Z๐ร๐รM๐ร{=,โฅ}, denote by ๐(t) thesolution set of the linear Diophantine system. Compute the generating function
โ๐ด,๐,โฆ(z,q) =โtโZ๐
โโ โxโ๐(t)
zx
โโ qt.
๏ฟฝ
Definition 2.3 (Counting Linear Diophantine Problem)Given a linear Diophantine system (๐ด, ๐,โฆ) โ Z๐ร๐รM๐ร{=,โฅ}, denote by ๐(t) thesolution set of the system. Compute the generating function
๐๐ด,๐,โฆ(t) =โtโZ๐
|๐(t)|qt.
๏ฟฝ
Complexity
The reason why we restrict to linear systems is that algorithmic treatment of the general(polynomial) problem is not feasible. This is the celebrated Hilbertโs 10th problem, whichMatiyasevich [32] solved in the negative direction. In other words, there is no algorithmto decide if a polynomial has integer solutions.
Although the restriction to linear systems makes the problem algorithmically solv-able, it does not make it efficiently solvable. The problem remains NP-hard.
Reduction
The decision problem of linear Diophantine systems (LDS) is
decide if there exists ๐ฅ โ N๐
such that ๐ด๐ฅ โฅ ๐
In order to show that LDS is NP-hard we reduce from 3-SAT [26]:
given a boolean expression ๐ต(๐ฅ) =โ๐
๐=1๐ถ๐ in ๐-variables,where ๐ถ๐ is a disjunction of three literalsdecide if there exists ๐ฅ โ {0, 1}๐ such that ๐ต(๐ฅ) = 1
2.1. INTRODUCTION 55
For each clause ๐๐ โจ๐๐ โจ๐๐, where ๐๐ข is either ๐ฆ๐ข or ยฌ๐ฆ๐ข, we construct an inequality๐๐ +๐๐ +๐๐ โฅ 1 where ๐๐ข is ๐ฅ๐ข or (1โ ๐ฅ๐ข) accordingly.
Let ๐๐๐ฅ โฅ ๐๐ be the inequality corresponding to the ๐-th clause of ๐ต(๐ฅ). Moreover,
let ๐ =
โกโขโขโขโฃ๐1๐2...๐๐
โคโฅโฅโฅโฆ โ Z๐ร๐ , ๐+ =
โกโขโขโขโฃ๐1๐2...๐๐
โคโฅโฅโฅโฆ โ Z๐ , ๐โ =
โกโขโขโขโฃโ1โ1...โ1
โคโฅโฅโฅโฆ โ Z๐ and ๐ผ๐ be the rank
๐ unit matrix.
Then if ๐ด =
[๐โ๐ผ๐
]and ๐ =
[๐+
๐โ
]we have
๐ฅ โ N๐ such that ๐ด๐ฅ โฅ ๐โ ๐ต(๐ฅ) = 1.
We note that this reduction is polynomial.
56 CHAPTER 2. LINEAR DIOPHANTINE SYSTEMS
2.2 Classification
We examine the hierarchy of problems related to linear Diophantine systems from theperspective of the nature of the problem. Although many problems admit the samealgorithmic solution, their nature may be very diverse. Similarly, problems that looksimilar may differ a lot when it comes to their algorithmic solution. There is a twofoldmotivation for this classification. On one hand, it clarifies (by refining) the specificationsof the linear Diophantine problems. On the other hand, it provides useful insight con-cerning the algorithmics, by providing base, degenerate or special cases for the โgeneralโproblems.
Moreover, since many problems are equivalent, it is useful to have an overview of thealternatives one has for solving a particular instance of a problem by reformulating it asan instance of another.
Another important aspect is that a problem (and its solution) may change a lotdepending on whether one is interested in the full generating function โ (listing problem)or the counting generating function ๐ (counting problem).
Each of these two types of problems contains several subproblems, depending on thedomain of the coefficients and the relation involved (equation or inequality).
2.2.1 Classification
For a problem (๐ด, ๐,โฆ), we will use a four letter notation MHRT, where M is the domainof the matrix ๐ด, H is M if the problem is parametric with respect to ๐, O if the problem ishomogeneous or the domain of ๐ if the problem is inhomogeneous, R is โฅ or = dependingon whether we have inequalities or equations and T is B for bounded or U for unboundedproblems.
In the diagram we see the inclusion relations (if a class A is included in class B, thenan instance from A is also an instance of B).
The following list gives some rules eliminating some of the cases:
1. *๐ *๐ต has {0} as its solution set.
2. N* โฅ * is unbounded.
3. N๐ * * is trivially satisfied (N๐ is its solution set).
4. NZ โฅ * is trivially satisfied (N๐ is its solution set).
5. NZ=* is infeasible.
We note that a problem is assumed to belong to one of the above classes and notany of its subclasses in order to apply the rules. For example, the last rule means thatthe inhomogeneous part has a negative entry, otherwise we would consider the problemin the class of problems with inhomogeneous part in N.
Omitting the cases covered above, we obtain the following diagram:
2.2. CLASSIFICATION 57
Z,M,=, ๐ Z,M,โฅ, ๐
N,M,โฅ, ๐
Z,M,=, ๐ต Z,M,โฅ, ๐ต
N,M,=, ๐ต
Z,Z,=, ๐ Z,Z,โฅ, ๐
Z,Z,=, ๐ต Z,Z,โฅ, ๐ต
Z,N,=, ๐ Z,N,โฅ, ๐
N,N,โฅ, ๐
Z,N,=, ๐ต
N,N,=, ๐ต
Z, 0,=, ๐ Z, 0,โฅ, ๐
We now present examples for some of the cases and provide names and referencesused in literature when available.
58 CHAPTER 2. LINEAR DIOPHANTINE SYSTEMS
N,N,=, ๐ต
Name: Magic Squares
References: [4], Ch.4. Prop 4.6.21 in [35], S6.2 in [17]
Description: 3ร 3 symmetric matrices of row and column sum equal to ๐
Matrix ๐ด vector ๐ Parametersโโโโโ1 1 1 0 0 0
0 1 0 1 1 0
0 0 1 0 1 1
โโโโโ โโโโโ
1
1
1
โโโโโ ๐
Listing (๐ง0๐ง1๐ง2๐ง3๐ง4๐ง5๐3โ1)(๐ง1๐ง5๐โ1)(๐ง0๐ง4๐โ1)(๐ง2๐ง3๐โ1)(๐ง0๐ง3๐ง5๐โ1)(๐ง1๐ง2๐ง4๐2โ1)
Counting (1+๐+๐2
(1โ๐)4(1+๐)
Name: Magic Pentagrams
References: S3, page 14 in [5]
Description: Partitions of ๐ into parts satisfying relations given by a pentagram withthe parts as vertices and inequalities given by the edges.
Matrix ๐ด vector ๐โโโโโโโโโโโ
1 0 0 1 0 1 0 0 0 1
1 0 1 0 0 0 1 1 0 0
0 1 0 0 1 1 1 0 0 0
0 0 1 0 1 0 0 0 1 1
0 1 0 1 0 0 0 1 1 0
โโโโโโโโโโโ
โโโโโโโโโโโ
2
2
2
2
2
โโโโโโโโโโโ Listing
๐๐ง210๐ง23๐ง
37๐ง
34
(โ๐ง3๐ง6+๐ง7๐ง10)(โ๐ง3๐ง4+๐ง8๐ง10)(๐ง4๐ง27๐ง9๐ง10๐โ1)(๐ง2๐ง3๐ง4๐ง7๐ง10๐โ1)ยท
1(๐ง3๐ง24๐ง5๐ง7๐โ1)(๐ง3๐ง24๐ง
27๐ง10๐โ๐ง1)
Counting (๐2+3๐+1)(๐2+13๐+1)(๐โ1)6
2.2. CLASSIFICATION 59
Z,Z,โฅ, ๐ต
Name: Solid Partitions on a Cube
References: S4.1 in [4]
Description: Partitions of ๐ subject to relations given by a cube where the vertices arethe parts and the directed edges indicate the inequalities.
Matrix ๐ด vector ๐โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
1 โ1 0 0 0 0 0 0
1 0 โ1 0 0 0 0 0
1 0 0 0 โ1 0 0 0
0 1 0 โ1 0 0 0 0
0 1 0 0 0 โ1 0 0
0 0 1 โ1 0 0 0 0
0 0 1 0 0 0 โ1 0
0 0 0 1 0 0 0 โ1
0 0 0 0 1 โ1 0 0
0 0 0 0 1 0 โ1 0
0 0 0 0 0 1 0 โ1
0 0 0 0 0 0 1 โ1
1 1 1 1 1 1 1 1
โ1 โ1 โ1 โ1 โ1 โ1 โ1 โ1
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
โโโโโโโโโโโ
0...
0
1
โ1
โโโโโโโโโโโ
Listing ๐ง4๐ง2๐ง1๐3
(1โ๐ง6๐)(๐ง5๐โ1)(๐ง1๐โ1)(๐ง1๐ง2๐2โ1)(๐ง1๐ง2๐ง4๐3โ1)(๐ง1๐ง2๐ง3๐ง4๐4โ1)ยท
1(๐ง1๐ง2๐ง3๐ง4๐ง5๐ง7๐6โ1)(๐ง1๐ง2๐ง3๐ง4๐ง5๐ง6๐ง7๐ง8๐8โ1)
Counting (๐16+2๐14+2๐13+3๐12+3๐11+5๐10+4๐9+8๐8+4๐7+5๐6+3๐5+3๐4+2๐3+2๐2+1)(๐โ1)8(๐+1)4(๐2+1)2(๐2+๐+1)2(๐2โ๐+1)(๐4+1)(๐4+๐3+๐2+๐+1)
ยท1
(๐6+๐5+๐4+๐3+๐2+๐+1)
60 CHAPTER 2. LINEAR DIOPHANTINE SYSTEMS
N,M,=, ๐ต
Name: Vector Partition Function
References: Section 4 in [15]
Description: Non-negative integer counts the natural solutions of
๐ฅ1 + 2๐ฅ2 + ๐ฅ3 = ๐1
๐ฅ1 + ๐ฅ2 + ๐ฅ4 = ๐2
Matrix ๐ด vector ๐ Parametersโโ1 2 1 0
1 1 0 1
โโ โโ ๐1
๐2
โโ ๐1, ๐2
Counting
โงโชโชโชโจโชโชโชโฉ๐214 + ๐1 +
7+(โ1)๐1
8 , ๐1 โค ๐2
๐1๐2 โ๐214 โ
๐222 + ๐1+๐2
2 + 7+(โ1)๐1
8 , ๐1 > ๐2 >๐1โ32
๐222 โ
3ยท๐22 + 1 , ๐2 โค ๐1โ3
2
Z, 0,โฅ, ๐ต
Name: Lecture Hall Partitions
References: [18]
Description: Partitions of ๐ where 2๐ โค ๐, 3๐ โค 2๐ and 4๐ โค 3๐.
Matrix ๐ด vector ๐โโโโโ2 โ1 0 0
0 3 โ2 0
0 0 4 โ3
โโโโโ โโโโโ
0
0
0
โโโโโ Listing
โ๐ง1๐ง32๐ง43๐ง
54๐
13
(๐ง2๐ง3๐ง4๐3โ1)(๐ง1๐ง22๐ง23๐ง
24๐
7โ1)(๐ง1๐ง22๐ง33๐ง
34๐
9โ1)(๐ง1๐ง22๐ง33๐ง
44๐
10โ1)
Counting (๐2โ๐+1)(๐4+1)(๐โ1)4(๐2+๐+1)(๐4+๐3+๐2+๐+1)
2.2. CLASSIFICATION 61
Z,M,โฅ, ๐ต
Name: Vector Partition Function
References: Section 4 in [15]
Description: Non-negative integer solutions ofโ๐ฅ1 โ 2๐ฅ2 โฅ ๐1
โ๐ฅ1 โ ๐ฅ2 โฅ ๐2
Matrix ๐ด vector ๐ Parametersโโโ1 โ2
โ1 โ1
โโ โโ ๐1
๐2
โโ ๐1, ๐2
Counting
โงโชโชโชโจโชโชโชโฉ๐214 + ๐1 +
7+(โ1)๐1
8 , ๐1 โค ๐2
๐1๐2 โ๐214 โ
๐222 + ๐1+๐2
2 + 7+(โ1)๐1
8 , ๐1 > ๐2 >๐1โ32
๐222 โ
3ยท๐22 + 1 , ๐2 โค ๐1โ3
2
Z,Z,โฅ, ๐
Matrix ๐ด vector ๐โโ1 โ1
1 โ2
โโ โโ 3
โ2
โโ
0 2 4 6 8 10
0
2
4
6
8
Listingโ๐ง21
(๐ง1โ1)(โ๐ง21+๐)(๐ง21๐ง2โ1)
Counting 5๐(2๐+3)(๐โ1)3
62 CHAPTER 2. LINEAR DIOPHANTINE SYSTEMS
N,M,โฅ, ๐
Name: Vector Partition Function
Matrix ๐ด vector ๐โโ1 3
2 1
โโ โโ ๐1
๐2
โโ
0 2 4 6 8 10
0
2
4
6
8
0 2 4 6 8 10
0
2
4
6
8
Listing ๐(1โ๐ง1)(1โ๐ง2)(1โ๐1๐ง1)(1โ๐22๐ง1)(1โ๐1๐22๐ง1)(1โ๐31๐ง2)(1โ๐2๐ง2)(1โ๐31๐2๐ง2)where ๐ = ๐52๐ง
31๐ง
32๐
71 + ๐42๐ง
31๐ง
32๐
71 โ ๐42๐ง
31๐ง
22๐
71 โ ๐32๐ง
21๐ง
22๐
71 + ๐52๐ง
31๐ง
32๐
61 + ๐42๐ง
31๐ง
32๐
61 โ
๐52๐ง21๐ง
32๐
61 โ ๐42๐ง
21๐ง
32๐
61 โ ๐32๐ง
21๐ง
32๐
61 โ ๐22๐ง
21๐ง
32๐
61 โ ๐42๐ง
31๐ง
22๐
61 + ๐42๐ง
21๐ง
22๐
61 โ ๐32๐ง
21๐ง
22๐
61 +
๐22๐ง21๐ง
22๐
61 + ๐32๐ง1๐ง
22๐
61 + ๐2๐ง1๐ง
22๐
61 + ๐52๐ง
31๐ง
32๐
51 + ๐42๐ง
31๐ง
32๐
51 โ ๐52๐ง
21๐ง
32๐
51 โ ๐42๐ง
21๐ง
32๐
51 โ
๐32๐ง21๐ง
32๐
51 โ ๐22๐ง
21๐ง
32๐
51 + ๐32๐ง1๐ง
32๐
51 + ๐22๐ง1๐ง
32๐
51 โ ๐42๐ง
31๐ง
22๐
51 + ๐42๐ง
21๐ง
22๐
51 โ ๐32๐ง
21๐ง
22๐
51 +
๐22๐ง21๐ง
22๐
51 โ๐2๐ง
22๐
51 +๐32๐ง1๐ง
22๐
51 โ๐22๐ง1๐ง
22๐
51 +๐2๐ง1๐ง
22๐
51 โ๐42๐ง
31๐ง
22๐
41 โ๐52๐ง
21๐ง
22๐
41 โ๐42๐ง
21๐ง
22๐
41 โ
3๐32๐ง21๐ง
22๐
41 โ ๐22๐ง
21๐ง
22๐
41 โ ๐2๐ง
21๐ง
22๐
41 โ ๐2๐ง
22๐
41 + 2๐32๐ง1๐ง
22๐
41 + ๐22๐ง1๐ง
22๐
41 + 2๐2๐ง1๐ง
22๐
41 +
๐42๐ง21๐ง2๐
41 + 2๐32๐ง
21๐ง2๐
41 + 2๐22๐ง
21๐ง2๐
41 + ๐2๐ง
21๐ง2๐
41 โ ๐22๐ง1๐ง2๐
41 โ ๐2๐ง1๐ง2๐
41 โ ๐42๐ง
31๐ง
22๐
31 +
๐42๐ง21๐ง
22๐
31โ๐32๐ง
21๐ง
22๐
31+๐22๐ง
21๐ง
22๐
31โ๐2๐ง
22๐
31+2๐32๐ง1๐ง
22๐
31+๐22๐ง1๐ง
22๐
31+2๐2๐ง1๐ง
22๐
31+๐32๐ง
21๐ง2๐
31+
๐22๐ง21๐ง2๐
31โ๐32๐ง1๐ง2๐
31โ2๐22๐ง1๐ง2๐
31โ2๐2๐ง1๐ง2๐
31โ๐ง1๐ง2๐
31โ๐42๐ง
31๐ง
22๐
21+๐42๐ง
21๐ง
22๐
21โ๐32๐ง
21๐ง
22๐
21+
๐22๐ง21๐ง
22๐
21 โ ๐2๐ง
22๐
21 + ๐32๐ง1๐ง
22๐
21 โ ๐22๐ง1๐ง
22๐
21 + ๐2๐ง1๐ง
22๐
21 + ๐32๐ง
21๐ง2๐
21 + ๐22๐ง
21๐ง2๐
21 + ๐2๐ง2๐
21 โ
๐32๐ง1๐ง2๐21 โ ๐22๐ง1๐ง2๐
21 โ ๐2๐ง1๐ง2๐
21 โ๐ง1๐ง2๐
21 +๐ง2๐
21 โ ๐32๐ง
21๐1 โ ๐22๐ง
21๐1 โ ๐2๐ง
21๐1 โ ๐32๐ง
21๐ง
22๐1 โ
๐2๐ง22๐1 + ๐32๐ง1๐ง
22๐1 + ๐2๐ง1๐ง
22๐1 + ๐2๐ง1๐1 + ๐42๐ง
21๐ง2๐1 +2๐32๐ง
21๐ง2๐1 +2๐22๐ง
21๐ง2๐1 + ๐2๐ง
21๐ง2๐1 +
๐2๐ง2๐1 โ ๐32๐ง1๐ง2๐1 โ 2๐22๐ง1๐ง2๐1 โ 2๐2๐ง1๐ง2๐1 โ ๐ง1๐ง2๐1 + ๐ง2๐1 + ๐2๐ง1 โ ๐22๐ง1๐ง2 โ ๐2๐ง1๐ง2 +1
Chapter 3
Partition Analysis
3.1 Partition Analysis Revisited
Partition analysis is a general methodology for the treatment of linear Diophantinesystems. The methodology is commonly attributed to MacMahon [31], since he wasthe first to apply it in a way similar to the one used today, that is, for the solution ofcombinatorial problems subject to linear Diophantine systems.
MacMahonโs motivation was the proof of his conjectures about plane partitions. Thiswas his declared goal in Combinatory Analysis. The fact that he failed to actually givean answer to the problem was like signing the death certificate of his own child. Incombination with other reasons, including the lack of interest for computational proce-dures in mathematics for most of the 20th century, MacMahonโs method did not becomemainstream among mathematicians.
Except for the theoretical side of the method, there is an algorithmic aspect. Thecomputational and algorithmic nature of the method was evident since the beginning.MacMahon employed an algorithm of Elliott in order to turn his method into an algo-rithm. In [24], Elliott introduced the basics of the methodology and used it to solvelinear homogeneous Diophantine equations. In particular, Elliottโs method is of inter-est because it is algorithmic (in the strict modern sense of the term). Elliott himself,although lacking modern terminology, is arguing on the termination of the procedure.Naturally, given the lack of computers at that time, MacMahon resorted in findingshortcuts (rules) for the most usual cases. This list of rules, included in CombinatoryAnalysis, he expanded as needed for the problem at hand.
It was Andrews who observed the computational potential of MacMahonโs partitionanalysis and waited for the right problem to apply it. In the late 1990โs, the seminal pa-per of BousquetโMelou and Eriksson [18] on lecture-hall partitions appeared. Andrewssuggested to Paule to explore the capacity of partition analysis combined with sym-bolic computation. This collaboration gave a series of papers, among which [4] and [5]deal with implementations of two fully algorithmic versions of partition analysis, calledOmega, empowered by symbolic computation.
There are different algorithmic (and non-algorithmic) realizations of the general idea.
63
64 CHAPTER 3. PARTITION ANALYSIS
Their relation is illustrated in the following figure. In what follows we will see in chrono-logical order the milestones of partition analysis. Although the goal is to present thehistory, we will allow for a modern view. In particular, we will see some geometric andalgebraic aspects that help understanding partition analysis.
MacMahon
Xin
Ell Ell2
AndrewsPauleRiese
omega
omega2
Elliott
PolyhedralOmega
3.1.1 Elliott
One of the first references relevant to partition analysis is Elliottโs article โOn linearhomogeneous Diophantine equationโ [24]. The work of Elliott is exciting, if not foranything else, because it addresses mathematicians that lived a century apart. It wasof interest to MacMahon, who based his method on Elliottโs decomposition, but also to21st century mathematicians for explicitly giving an important algorithm. Although, aswe shall see, the algorithm has very bad complexity, it can be considered as an earlyalgorithm for the enumeration of lattice points in cones (among other things).
The problem Elliott considers is to find all non-negative integer solutions to theequation
๐โ๐=1
๐๐๐ฅ๐ โ๐+๐โ
๐=๐+1
๐๐๐ฅ๐ = 0 for ๐๐, ๐๐ โ N. (3.1)
In other words, we consider one homogeneous linear Diophantine equation. It should benoted that Elliott himself (as well as subsequent authors) expressed the equation in theform Diophantus would prefer, without using negative coefficients.
The starting point for Elliottโs work is the fact that even if one computes the set ofโsimple solutionsโ, i.e., solutions that are not combinations of others, there are syzygies
3.1. PARTITION ANALYSIS REVISITED 65
preventing us from writing down formulas giving each and every solution of the equationexactly once. He proceeds by explaining that his method computes a generating functionwhose terms are in one-to-one correspondence with the solutions of the equation. Elliottstates that this generating function is obtained โby a finite succession of simple stagesโ.This sentence exhibits an impressive insight on what an algorithm is, as well as Elliottโsrealization that he has an algorithm at hand. We quote Elliott outlining his idea in [24]:
The principle is that in the infinite expansion which is the formal product ofthe infinite expansions1 + ๐1๐ข
๐1 + ๐21๐ข2๐1 + ยท ยท ยท
1 + ๐2๐ข๐2 + ๐22๐ข
2๐2 + ยท ยท ยท...1 + ๐๐๐ข๐๐ + ๐2๐๐ข2๐๐ + ยท ยท ยท1 + ๐๐+1๐ข
โ๐๐+1 + ๐2๐+1๐ขโ2๐๐+1 + ยท ยท ยท
1 + ๐๐+2๐ขโ๐๐+2 + ๐2๐+2๐ข
โ2๐๐+2 + ยท ยท ยท...1 + ๐๐+๐๐ข
โ๐๐+๐ + ๐2๐+๐๐ขโ2๐๐+๐ + ยท ยท ยท
the terms free from ๐ข are of the form ๐๐ฅ11 ๐๐ฅ2
2 ยท ยท ยท ๐๐ฅ๐๐ ๐
๐ฅ๐+1
๐+1 ๐๐ฅ๐+2
๐+2 ยท ยท ยท ๐๐ฅ๐+๐๐+๐ with
1 for numerical coefficient, where ๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐, ๐ฅ๐+1, . . . , ๐ฅ๐+๐ are a set ofpositive integers (zero included), which satisfy our Diophantine equation 3.1,and that there is just one such term for each set of solutions.
In other words, the generating function for determining the sets of solu-tions, each once, of 3.1, as sets of exponents of the ๐๐โs in its several terms,is the expression for the part which is free from ๐ข of the expansion of
1
(1โ ๐1๐ข๐1) ยท ยท ยท (1โ ๐๐๐ข๐๐)(1โ ๐๐+1๐ขโ๐๐+1) ยท ยท ยท (1โ ๐๐+๐๐ขโ๐๐+๐)(3.2)
in the positive powers of the ๐๐โs.The problem is, in any case, to extract from 3.2, and to examine, this
generating function. The extraction may be effected by a finite number ofeasy steps as follows.
The principle presented here by Elliott is the basis of partition analysis, i.e., intro-ducing an extra variable, denoted by ๐ข in Elliott and by ๐ in modern partition analysis.We switch from ๐ and ๐ข to ๐ง and ๐ in order to conform with more modern notationalconventions. The method of Elliott computes a partial fraction decomposition of anexpression
๐โ๐
1
1โ๐๐, (3.3)
where ๐๐ โ[๐ง1, ๐ง2, . . . , ๐ง๐, ๐, ๐
โ1]and ๐ โ N, into a sum of the formโ ยฑ1โ(1โ ๐๐)
+โ ยฑ1โ
(1โ ๐๐)(3.4)
66 CHAPTER 3. PARTITION ANALYSIS
with ๐๐ โ [๐ง1, ๐ง2, . . . , ๐ง๐, ๐] and ๐๐ โ [๐ง1, ๐ง2, . . . , ๐ง๐, ๐โ1].
We note that some authors refer to rational functions of the form 3.3 as Elliott rationalfunctions. The algorithm is based on the fact
1
(1โ ๐ฅ๐๐ผ)(1โ ๐ฆ๐๐ฝ )
=1
1โ ๐ฅ๐ฆ๐๐ผโ๐ฝ
(1
1โ ๐ฅ๐๐ผ+
1
1โ ๐ฆ๐๐ฝ
โ 1
)(3.5)
=1
(1โ ๐ฅ๐ฆ๐๐ผโ๐ฝ)(1โ ๐ฅ๐๐ผ)+
1
(1โ ๐ฅ๐ฆ๐๐ผโ๐ฝ)(1โ ๐ฆ๐๐ฝ )โ 1
(1โ ๐ฅ๐ฆ๐๐ผโ๐ฝ).
Observe that given ๐ผ and ๐ฝ positive integers, after applying 3.5, we obtain a sum ofterms where in each of them either the number of factors containing ๐ reduced or theexponent of ๐ reduced (in absolute value) in one of the factors while it did not changein the other. This observation is the proof of termination Elliott gives for his algorithm.
Letโs see a very simple example, where by applying (3.5) once, we obtain the desiredpartial fraction decomposition.
Example 7.
1
(1โ ๐ฅ๐)(1โ ๐ฆ๐)
=1
(1โ ๐ฅ๐ฆ)(1โ ๐ฅ๐)+
1
(1โ ๐ฅ๐ฆ)(1โ ๐ฆ๐)โ 1
(1โ ๐ฅ๐ฆ)
=
(1
(1โ ๐ฅ๐ฆ)(1โ ๐ฅ๐)โ 1
(1โ ๐ฅ๐ฆ)
)+
1
(1โ ๐ฅ๐ฆ)(1โ ๐ฆ๐)
.
Notice that the terms in the parenthesis contain only non-negative ๐ exponents, whilethe last term contains only non-positive ones as requested by (3.4). ๏ฟฝ
Each of the summands inโ๐
ยฑ1โ๐(1โ ๐๐)
can be expanded using the geometric-series
expansion formula. It is easy to see that the summands contributing to the generatingfunction for the solution of Equation 3.1 are exactly the ones where in their expansion
๐ does not appear. This happens only for the termsยฑ1โ
๐(1โ ๐๐)where all ๐๐ are ๐-free.
Summing up only these terms we get the desired generating function.In order to translate this rational function identity to an identity about cones we
first observe that1
(1โ ๐ฅ๐๐ผ)(1โ ๐ฆ๐๐ฝ )
is the generating function of the 2-dimensional cone ๐ถ = ๐R ((1, 0, ๐ผ), (1, 0,โ๐ฝ)), sinceLemma 2 guarantees that the numerator in the rational generating function of ๐ถ isindeed 1.
Observe that the point (1, 1, ๐ผโ ๐ฝ) is in the interior of the cone ๐ถ. This means thatthe cones ๐ด = ๐R ((1, 0, ๐ผ), (1, 1, ๐ผโ ๐ฝ)) and ๐ต = ๐R ((1, 0,โ๐ฝ), (1, 1, ๐ผโ ๐ฝ)) subdividethe cone ๐ถ. Their intersection is exactly the ray starting from the origin and passing
3.1. PARTITION ANALYSIS REVISITED 67
through (1, 1, ๐ผ โ ๐ฝ). By a simple inclusion-exclusion argument we have the signeddecomposition ๐ถ = ๐ด+ ๐ต โ (๐ด โฉ ๐ต). This decomposition translated to the generatingfunction level is exactly the partial fraction decomposition employed by Elliott. In theexample shown below, after three applications of Elliottโs decomposition we obtain thedesired cone.
๐ฅ ๐ฆ
๐
๐ฅ ๐ฆ
๐
๐ฅ ๐ฆ
๐
๐ฅ ๐ฆ
๐
๐ฅ ๐ฆ
๐
๐ฅ ๐ฆ
๐
๐ฅ ๐ฆ
๐
๐ฅ ๐ฆ
๐
Elliottโs decomposition
The Algorithm
It is easy to see that, after a finite number of steps, we end up with a sum of cones ofthree types:
1. the generators contain only zero last coordinate (๐-coordinate);
2. the generators contain only non-negative (but not all zero) last coordinate;
3. the generators contain only non-positive (but not all zero) last coordinate.
Note that all the cones involved are unimodular.For equations we sum up the generating functions corresponding to cones of the 1st
type. For inequalities, we intersect each cone with the non-negative ๐ halfspace. Thismeans that we discard the cones of the 3rd type and sum up the generating functions ofthe rest of the cones. Then we (orthogonally) project with respect to the ๐-coordinate.The sum of rational generating functions for the lattice points in each cone we obtainedis the partial fraction decomposition obtained by Elliottโs algorithm.
68 CHAPTER 3. PARTITION ANALYSIS
Elliott for Systems
It is important to note that Elliottโs method relies on the fact that the numerators inall the expressions involved are equal to 1. Of course after bringing all the terms thealgorithm returns over a common denominator, there is no guarantee that the numeratorwill be 1. But for each term returned the condition is preserved. Thus it is possible toiteratively apply the algorithm to eliminate ๐โs in order to solve systems of equationsor inequalities. We note that Elliott proves that the numerators will always be ยฑ1. Norepetitions occur in the computed expression.
Example 8. The system of linear homogeneous inequalities {7๐ โ ๐ โฅ 0, ๐ โ 3๐ โฅ 0}can be solved by Elliott, resulting in 1
(1โ๐ฅ)(1โ๐ฅ3๐ฆ), if the inequality ๐ โ 3๐ โฅ 0 is treated
first. If the order is reversed then the intermediate expression
โ1โ ๐ฅ๐ฆ6
๐17 โ ๐ฅ๐ฆ5
๐14 โ ๐ฅ๐ฆ4
๐11 โ ๐ฅ๐ฆ3
๐8 โ ๐ฅ๐ฆ2
๐5 โ ๐ฅ๐ฆ๐2
(1โ ๐ฅ๐)(1โ ๐ฅ๐ฆ7
๐20 )
is violating the condition that numerators are equal to 1.The first inequality of the system is redundant, i.e., if we ignore it the solution set does
not change. Thus, choosing the right order, the intermediate expression 1(1โ๐ฅ๐7)(1โ๐ฅ3๐ฆ๐20)
behaves well. We could still apply Elliottโs method if we did not bring the intermediateexpression over a common denominator. ๏ฟฝ
3.1.2 MacMahon
MacMahon presented his investigations concerning integer partition theory in a series ofseven memoirs, published between 1895 and 1916. In the second memoir [30], MacMahonobserves that the theory of partitions of numbers develops in parallel to that of linearDiophantine equations and in his masterpiece โCombinatory Analysisโ [31] he notes:
The most important part of the volume. Sections VIII et seq., arises frombasing the theory of partitions upon the theory of Diophantine inequalities.This method is more fundamental than that of Euler, and leads directly to ahigh degree of generalization of the theory of partitions, and to several inves-tigations which are grouped together under the title of โPartition Analysisโ.
In order to attack the problem of solving linear Diophantine systems, MacMahon(like Elliott) introduced extra variables. Let ๐ด โ Z๐ร๐ and b โ Z๐. We want to findall x โ N๐ satisfying ๐ดx โฅ b. The generating function of the solution set is
ฮฆ๐ด,๐(z) =โ๐ดxโฅb
zx.
Letโs introduce extra variables ๐1, ๐2, . . . , ๐๐ to encode the inequalities ๐ดx โฅ b, trans-forming the generating function to โ
xโN๐,๐ดxโฅb
๐๐ดxzx
3.1. PARTITION ANALYSIS REVISITED 69
The ๐ variables are introduced to encode the inequalities, but the solutions we aresearching for live in dimension ๐. The ๐ dimensions are used to control that aftera certain decomposition we can filter out the solutions involving negative exponents.Following this principle, MacMahon introduced the concept of the crude generatingfunction. But before that, we need to see the ฮฉโฅ operator. MacMahon defined it inArticle 66 of [30]:
Suppose we have a function
๐น (๐ฅ, ๐) (3.6)
which can be expanded in ascending powers of ๐ฅ. Such expansion being eitherfinite or infinite, the coefficients of the various powers of ๐ฅ are functions of๐ which in general involve both positive and negative powers of ๐. We mayreject all terms containing negative powers of ๐ and subsequently put ๐ equalto unity. We thus arrive at a function of ๐ฅ only, which may be representedafter Cayley (modified by the association with the symbol โฅ) by
ฮฉโฅ ๐น (๐ฅ, ๐) (3.7)
the symbol โฅ denoting that the terms retained are those in which the powerof ๐ is โฅ 0.
The notation ฮฉโฅ is now a standard and usually referred to as โMacMahonโs Omegaโ. Itis worth noting, at least in order to give credit where credit is due, as MacMahoon did,that the first use of this notation is by Cayley in [22]. MacMahon extends the notationto multivariate functions and defines the ฮฉ= operator.
A more modern definition of the ฮฉโฅ operator is given in [4], by Andrews,Paule andRiese:
The ฮฉโฅ operator is defined on functions with absolutely convergent mul-tisum expansions
โโ๐ 1=โโ
โโ๐ 2=โโ
ยท ยท ยทโโ
๐ ๐=โโ๐ด๐ 1,๐ 2,...,๐ ๐๐
๐ 11 ๐๐ 2
2 ยท ยท ยท๐๐ ๐๐
in an open neighborhood of the complex circles |๐๐| = 1. The action of ฮฉโฅis given by
ฮฉโฅ
โโ๐ 1=โโ
โโ๐ 2=โโ
ยท ยท ยทโโ
๐ ๐=โโ๐ด๐ 1,๐ 2,...,๐ ๐๐
๐ 11 ๐๐ 2
2 ยท ยท ยท๐๐ ๐๐ :=
โโ๐ 1=0
โโ๐ 2=0
ยท ยท ยทโโ
๐ ๐=0
๐ด๐ 1,๐ 2,...,๐ ๐ .
70 CHAPTER 3. PARTITION ANALYSIS
This definition is concerned with convergence issues, while MacMahon considered ฮฉโฅacting on purely formal objects. In what follows we take the purely formal side (beingon safe ground due to geometry). But we also address the justified concerns aboutconvergence, relating them to the geometric issues.
In order to use the ฮฉโฅ operator for the solution of linear Diophantine systems,MacMahon resorts to the construction of a generating function he calls crude. This isin principal a multivariate version of Elliottโs construction.
Definition 3.1 (Crude Generating Function)Given ๐ด โ Z๐ร๐ and b โ Z๐ we define the crude generating function as
ฮฆฮฉ๐ด,๐ (z;๐) := ฮฉโฅ
โxโN๐
zx๐โ๐=1
๐๐ด๐xโ๐๐๐ .
๏ฟฝ
We stress the fact that the assignment is meant formally, since it is easy to see that
ฮฉโฅฮฆ๐๐ด,๐(๐ง;๐) = ฮฆ๐ด,๐(z)
as well as
ฮฆฮฉ๐ด,๐(z;๐) =
โxโN๐,๐ดxโฅ๐
zx = ฮฆ๐ด,๐(z).
This means that ฮฉโฅโ
xโN๐ zxโ๐
๐=1 ๐๐ด๐xโ๐๐๐ is the answer to the problem of solving a
linear Diophantine system. Thus, it is expected that the computation of the action ofthe ฮฉโฅ operator is not easy.
We define the ๐-generating function as an intermediate step, both in order to increaseclarity in this section and because it is essential when discussing geometry later. Inprincipal, the ๐-generating function is the crude generating function without prependingฮฉโฅ. Given ๐ด โ Z๐ร๐ and b โ Z๐ we define the ๐ generating function as
ฮฆ๐๐ด,๐(z, ๐) =
โxโN๐
zx๐โ๐=1
๐๐ด๐xโ๐๐๐ .
Based on the geometric series expansion formula
(1โ ๐ง)โ1 =โ๐ฅโฅ0
๐ง๐ฅ
we can transform the series into a rational function. The rational form of ฮฆ๐๐ด,๐(z) is
denoted by ๐๐๐ด,๐(z) and it has the form
๐๐๐ด,๐(z, ๐) = ๐โb๐โ๐=1
1(1โ ๐ง๐๐(๐ด๐ )๐
)
3.1. PARTITION ANALYSIS REVISITED 71
We present an example that we will also use for the geometric interpretation of ฮฉโฅ,
Let ๐ด = [ 2 3 ] and ๐ = 5.
Then
ฮฆ๐๐ด,๐(๐ง1, ๐ง2, ๐) =
โ๐ฅ1,๐ฅ2โN
๐2๐ฅ1+3๐ฅ2โ5๐ง๐ฅ11 ๐ง๐ฅ2
2
and
๐๐๐ด,๐(๐ง1, ๐ง2, ๐) =๐โ5
(1โ ๐ง1๐2)(1โ ๐ง2๐3)
These generating functions can be directly translated to cones by examining theirrational form and recalling the form for the generating function of lattice points in cones.
In the following figure, we see the geometric steps for creating the ๐-cone (equivalentof the ๐-generating function) and then applying the ฮฉโฅ operator.
At first we lift the standard generators of the positive quadrant of R๐ by appendingthe exponents of the ๐ variables (thus lifting the generators in R๐+๐). We translate thelifted cone by the exponent of ๐ in the numerator. Then we intersect with the positivequadrant of R๐+๐ and project the intersection to R๐. The generating function of theobtained polyhedron is the result of the action of ฮฉโฅ on the ๐-generating function.
๐ฅ๐ฆ
๐
Consider the first quadrant and lift thestandard generators according to the
input.
๐ฅ๐ฆ
๐
Shift according to the inhomogeneouspart in the negative ๐ direction the
cone generated by the lifted generators.Intersect with the ๐ฅ๐ฆ-plane.
72 CHAPTER 3. PARTITION ANALYSIS
๐ฅ๐ฆ
๐
Take the part of the cone (apolyhedron) living above the ๐ฅ๐ฆ-plane
and project its lattice points.
๐ฅ๐ฆ
๐
The projected polyhedron contains thesolutions to the inequality.
Given the geometric interpretation of the crude generating function, one could arguethat it is actually a refined one. Or if we consider the generating series as a cloth-hanging rope, then the crude generating function is a cloth-hanging grid allowing us toparametrize the hanging.
3.1.3 MacMahonโs Rules
Although the machinery of MacMahon was very powerful, it was not very practical.Lacking computing machines, he had to resort to lookup tables in order to solve problemshe was interested in. In his partition theory memoirs and in โCombinatory Analysisโ,he presents a set of rules. These are adhoc rules he was inventing when needed in orderto solve some combinatorial problem he was presenting.
Here we present nine of the rules MacMahon used for the evaluation of the ฮฉโฅoperator, taken from [4]. Most of the proofs are easy, so we restrict to observationsusing geometric insight.
MacMahon Rules 1 & 2
The first two rules in MacMahonโs list are:For ๐ โ N* :
ฮฉโฅ1
(1โ ๐๐ฅ)(1โ ๐ฆ
๐๐
) =1
(1โ ๐ฅ) (1โ ๐ฅ๐ ๐ฆ)
and
ฮฉโฅ1
(1โ ๐๐ ๐ฅ)(1โ ๐ฆ
๐
) =1 + ๐ฅ๐ฆ 1โ๐ฆ๐ โ1
1โ๐ฆ
(1โ ๐ฅ) (1โ ๐ฅ๐ฆ๐ ).
3.1. PARTITION ANALYSIS REVISITED 73
The rules are about evaluating ฮฉโฅ1
(1โ๐๐ฅ)(1โ ๐ฆ๐๐ )
and ฮฉโฅ1
(1โ๐๐ ๐ฅ)(1โ ๐ฆ๐)
. There is an appar-
ent symmetry in the input, but elimination results in structurally different numerators,
i.e., 1 versus 1 + ๐ฅ๐ฆ 1โ๐ฆ๐ โ1
1โ๐ฆ . This phenomenon is better understood if we look at thesaturated semigroup that is the solution set of each linear Diophantine system. Thesetwo cases correspond to the inequalities ๐ฅ โ ๐ ๐ฆ โฅ 0 and ๐ ๐ฅ โ ๐ฆ โฅ 0 respectively. Thetwo cones generated by following the Cayley Lifting construction are
๐ฅ๐ฆ
๐
๐ฅ
๐ฆ
๐
๐ฅ
๐ฆ
๐
๐ฅ๐ฆ
๐
๐ฅ๐ฆ
๐
๐ฅ๐ฆ
๐
The geometry of the first two MacMahon rules.The numerator in the rational generating function expresses the lattice points in thefundamental parallelepiped. Since the first cone is unimodular, the numerator is 1. Forthe second cone, we observe that the fundamental parallelepiped, except for the origin,contains a vertical segment (truncated geometric series) at ๐ฅ = 1. The length of thesegment is ๐ and since the fundamental parallelepiped is half-open, the lattice points ofthis segment are (1, ๐) for ๐ = 1, 2, . . . , ๐ . The rational function
๐ฅ๐ฆ1โ ๐ฆ๐ โ1
1โ ๐ฆ=
๐ฅ๐ฆ โ ๐ฅ๐ฆ๐
1โ ๐ฆ
is exactly the generating function of that segment.
MacMahon Rule 4
ฮฉโฅ1
(1โ ๐๐ฅ) (1โ ๐๐ฆ)(1โ ๐ง
๐
) =1โ ๐ฅ๐ฆ๐ง
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง).
Although the rational function on which ฮฉโฅ acts has three factors in the denominator,the resulting rational generating function has four factors in the denominator. Moreover,we observe that the numerator has terms with both positive and negative signs. We willgive a geometric view on these observations. The ๐-cone defined in the left hand sideof the MacMahon rule is ๐R ((1, 0, 0, 1), (0, 1, 0, 1), (0, 0,โ1, 1)) . If we compute theintersection of this cone with the positive quadrant of R4, we obtain a cone generated by
74 CHAPTER 3. PARTITION ANALYSIS
(1, 0, 0, 1), (0, 1, 0, 1), (1, 0, 1, 0) and (0, 1, 1, 0). Letโs project this into R3 (by eliminatingthe last coordinate). We obtain a non-simplicial cone. We provide two decompositionsof the cone into simplicial cones. We used Normaliz [21] for the computations.
The first decomposition is by triangulation as shown here:
๐ฅ๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
The generating function of the green cone ๐R ((1, 0, 0), (0, 1, 0), (1, 0, 1)) is
1
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ฅ๐ง)
while that of the red cone ๐R ((0, 1, 0), (0, 1, 1), (1, 0, 1)) is
1
(1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง).
If we perform inclusion-exclusion (subtracting the purple cone), we obtain:
1
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ฅ๐ง)+
1
(1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)โ 1
(1โ ๐ฆ) (1โ ๐ฅ๐ง)=
(1โ ๐ฅ๐ฆ) (1โ ๐ฆ)
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)โ 1
(1โ ๐ฆ) (1โ ๐ฅ๐ง)=
(1โ ๐ฅ๐ฆ) (1โ ๐ฆ)โ (1โ ๐ฅ๐ง)
(1โ ๐ฆ) (1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)=
1โ ๐ฅ๐ฆ๐ง
(1โ ๐ฆ) (1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)
The second decomposition is not a triangulation, but a signed cone decomposition (a laBarvinok).
๐ฅ๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
We first consider the whole positive quadrant and then subtract the half-open cone
๐(0,0,1)R ((1, 0, 1), (0, 1, 1), (0, 0, 1)) . Half-opening the cone is equivalent (as far as latticepoints are concerned) to shifting the cone by the generator (0, 0, 1). Thus we have atthe generating function level:
3.1. PARTITION ANALYSIS REVISITED 75
1
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ง)โ ๐ง
(1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง) (1โ ๐ง)=
(1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)โ ๐ง (1โ ๐ฅ) (1โ ๐ฆ)
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ง) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)=
(1โ ๐ง)โ ๐ฅ๐ฆ๐ง (1โ ๐ง)
(1โ ๐ฆ) (1โ ๐ฆ) (1โ ๐ง) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)=
1โ ๐ฅ๐ฆ๐ง
(1โ ๐ฆ) (1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฆ๐ง)
MacMahon Rule 6
ฮฉโฅ1
(1โ ๐2๐ฅ)(1โ ๐ฆ
๐
) (1โ ๐ง
๐
) =1 + ๐ฅ๐ฆ + ๐ฅ๐ง + ๐ฅ๐ฆ๐ง
(1โ ๐ฅ) (1โ ๐ฅ๐ฆ2) (1โ ๐ฅ๐ง2).
In this rule we observe that we have a positive sum of four terms in the numerator.The ฮฉ-polyhedron is a cone that is simplicial but not unimodular. Observe in the figurethe four points in the fundamental parallelepiped, corresponding to the four terms in
the sum.
๐ฅ
๐ฆ
๐ง
๐ฅ
๐ฆ
๐ง
Other MacMahon Rules
Although the list of rules is not comprehensive, in the sense that they cannot cover allcases needed to treat linear Diophantine systems, they can solve certain combinatorialproblems. For the rest of MacMahonโs rules similar geometric observations apply, butthe geometry becomes more complicated and the reasoning behind the numerators orthe denominators appearing is not as simple anymore. A list of other MacMahon rulesis given here:
MacMahon Rule 3
ฮฉโฅ1
(1โ ๐๐ฅ)(1โ ๐ฆ
๐
) (1โ ๐ง
๐
) =1
(1โ ๐ฅ) (1โ ๐ฅ๐ฆ) (1โ ๐ฅ๐ง).
76 CHAPTER 3. PARTITION ANALYSIS
MacMahon Rule 5
ฮฉโฅ1
(1โ ๐๐ฅ) (1โ ๐๐ฆ)(1โ ๐ง
๐2
) =1 + ๐ฅ๐ฆ๐ง โ ๐ฅ2๐ฆ๐ง โ ๐ฅ๐ฆ2๐ง
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ฅ2๐ง) (1โ ๐ฆ2๐ง).
MacMahon Rule 7
ฮฉโฅ1
(1โ ๐2๐ฅ) (1โ ๐๐ฆ)(1โ ๐ง
๐
) =1 + ๐ฅ๐ง โ ๐ฅ๐ฆ๐ง โ ๐ฅ๐ฆ๐ง2
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ฆ๐ง) (1โ ๐ฅ๐ง2).
MacMahon Rule 8
1
(1โ ๐๐ฅ) (1โ ๐๐ฆ) (1โ ๐๐ง)(1โ ๐ค
๐
) =
1โ ๐ฅ๐ฆ๐ค โ ๐ฅ๐ง๐ค โ ๐ฆ๐ง๐ค + ๐ฅ๐ฆ๐ง๐ค + ๐ฅ๐ฆ๐ง๐ค2
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ง) (1โ ๐ฅ๐ค) (1โ ๐ฆ๐ค) (1โ ๐ง๐ค).
MacMahon Rule 9
ฮฉโฅ1
(1โ ๐๐ฅ) (1โ ๐๐ฆ)(1โ ๐ง
๐
) (1โ ๐ค
๐
) =
1โ ๐ฅ๐ฆ๐ง โ ๐ฅ๐ฆ๐ค โ ๐ฅ๐ฆ๐ง๐ค + ๐ฅ๐ฆ2๐ง๐ค + ๐ฅ2๐ฆ๐ง๐ค
(1โ ๐ฅ) (1โ ๐ฆ) (1โ ๐ฅ๐ง) (1โ ๐ฅ๐ค) (1โ ๐ฆ๐ง) (1โ ๐ฆ๐ค).
3.1.4 Andrews-Paule-Riese
In 1997, BousquetโMelou and Eriksson presented a theorem on lecture-hall partitionsin [18]. This theorem gathered a lot of attention from the community (and it stilldoes, with many lecture-hall type theorems appearing still today). Andrews, who hadalready studied MacMahonโs method and was aware of its computational potential,figured that lecture-hall partitions offered the right problems to attack algorithmicallyvia partition analysis. At the same time he planned to spend a semester during hissabbatical at the Research Institute for Symbolic Computation (RISC) in Austria towork with Paule. It is only natural that the result was a fully algorithmic versionof MacMahonโs method powered by symbolic computation. This collaboration gavea series of 10 papers [4, 1, 10, 5, 6, 7, 3, 8, 9, 2]. Many interesting theorems anddifferent kinds of partitions are defined and explored in this series, but for us the twomost important references are [4] and [5], which contain the algorithmic improvementson partition analysis. Namely, in [4] the authors introduce Omega, a Mathematicapackage based on a fully algorithmic partition analysis version, while in [5] they presenta more advanced partial fraction decomposition (and the related Mathematica packageOmega2) solving some of the problems appearing in Omega. While presenting the twomethods, we will see some geometric aspects of theirs.
3.1. PARTITION ANALYSIS REVISITED 77
Omega
The main tool in Omega is the Fundamental Recurrence for the ฮฉโฅ operator, given byLemma 6. Following MacMahon, or Elliott for that matter, iterative application of thisrecurrence is enough for computing the action of ฮฉโฅ.
Lemma 6 (Fundamental Recurrence, Theorem 2.1 in [4]). For ๐,๐ โ N* and ๐ โ Z:
ฮฉโฅ๐๐
(1โ ๐ฅ1๐) ยท ยท ยท (1โ ๐ฅ๐๐)(1โ ๐ฆ1๐ ) ยท ยท ยท (1โ ๐ฆ๐
๐ )=
๐๐,๐,๐(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐; ๐ฆ1, ๐ฆ2, . . . , ๐ฆ๐)โ๐๐=1(1โ ๐ฅ๐)
โ๐๐=1
โ๐๐=1(1โ ๐ฅ๐๐ฆ๐)
where for ๐ > 1
๐๐,๐,๐(๐ฅ1, . . . , ๐ฅ๐; ๐ฆ1, . . . , ๐ฆ๐) =
1
๐ฅ๐ โ ๐ฅ๐โ1
โโ๐ฅ๐(1โ ๐ฅ๐โ1)
๐โ๐=1
(1โ ๐ฅ๐โ1๐ฆ๐)๐๐โ1,๐,๐(๐ฅ1, . . . , ๐ฅ๐โ2, ๐ฅ๐; ๐ฆ1, . . . , ๐ฆ๐)
โ๐ฅ๐โ1(1โ ๐ฅ๐)๐โ๐=1
(1โ ๐ฅ๐๐ฆ๐)๐๐โ1,๐,๐(๐ฅ1, . . . , ๐ฅ๐โ2, ๐ฅ๐โ1; ๐ฆ1, . . . , ๐ฆ๐)
โโ and for ๐ = 1
๐1,๐,๐(๐ฅ1; ๐ฆ1, . . . , ๐ฆ๐) =
โงโชโจโชโฉ๐ฅโ๐1 if ๐ โค 0
๐ฅโ๐1 +
๐โ๐=1
(1โ ๐ฅ1๐ฆ๐)๐โ
๐=0
โ๐(๐ฆ1, . . . , ๐ฆ๐)(1โ ๐ฅ๐โ๐1 ) if ๐ > 0
๏ฟฝ
The recurrence looks intimidating, but bear in mind that it is supposed to be a computa-tional tool used in combination with symbolic computation. It is this power that earlierauthors did not have. Even the ones with extreme computational (by hand) abilities,like MacMahon, had to resort to lookup tables and sets of rules for the more usual cases.
The base cases for the recurrence are when either all the terms have positive ๐-exponents or all the terms have negative-๐ exponents. For the two base cases we recall thedefinition of complete homogeneous symmetric polynomials and some related notation.
Definition 3.2 (Complete Homogeneous Symmetric Polynomials, see [5])We define โ๐(๐ง1, ๐ง2, . . . , ๐ง๐) through the generating function
โโ๐=0
โ๐(๐ง1, ๐ง2, . . . , ๐ง๐)๐ก๐ =
1
(1โ ๐ง1๐ก)(1โ ๐ง2๐ก) ยท ยท ยท (1โ ๐ง๐๐ก).
What is needed for the base cases of the fundamental recurrence, is the partial sum ofthis generating function (series). The following definition is useful to set notation.
78 CHAPTER 3. PARTITION ANALYSIS
For ๐ โ Z, let
๐ป๐(๐ง1, ๐ง2, . . . , ๐ง๐) =
{โ๐๐=0 โ๐(๐ง1, ๐ง2, . . . , ๐ง๐) if ๐ โฅ 0
0 if ๐ < 0.
๏ฟฝ
Now, returning to [4], we have the two base cases:
Lemma 7 (Lemma 2.1 in [4]). For any integer ๐,
ฮฉโฅ๐๐ผ
(1โ ๐ฅ1๐)(1โ ๐ฅ2๐) . . . (1โ ๐ฅ๐๐)= ฮฉโฅ
โโ๐=0
โ๐(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐)๐๐+๐
= 1(1โ๐ฅ1)(1โ๐ฅ2)...(1โ๐ฅ๐)
โ๐ปโ๐โ1(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐).๏ฟฝ
Lemma 8 (Lemma 2.2 in [4]). For any integer ๐,
ฮฉโฅ๐๐ผ
(1โ ๐ฆ1๐ )(1โ ๐ฆ2
๐ ) . . . (1โ ๐ฆ๐๐ )
= ฮฉโฅ
โโ๐=0
โ๐(๐ฆ1, ๐ฆ2, . . . , ๐ฆ๐)๐๐โ๐
= ๐ป๐(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐). ๏ฟฝ
The fundamental recurrence in Lemma 6 assumes that the exponents of ๐ in thedenominator are ยฑ1. This is not a strong assumption, as noted in [4], since we canalways employ the following decomposition.
(1โ ๐ฅ๐๐) =
๐โ1โ๐=0
(1โ ๐๐๐ฅ1๐๐)
(1โ ๐ฆ
๐๐ ) =
๐ โ1โ๐=0
(1โ ๐๐๐ฆ1๐
๐)
where ๐ = ๐2๐๐๐ and ๐ = ๐
2๐๐๐ . The obvious drawback of this approach is that we introduce
complex coefficients instead of just ยฑ1, which were the only possible coefficients beforethe decomposition. This motivates the desire for a better recurrence.
Omega2
In [5] the authors introduce an improved partial fraction decomposition method, givenby the recurrence of Theorem 3.1.
Theorem 3.1 (Generalized Partial Fraction Decomposition, see [5])Let ๐ผ โฅ ๐ฝ โฅ 1, gcd(๐ผ, ๐ฝ) = 1 and let rmd(๐, ๐) denote the division remainder of ๐ by ๐.Then
1
(1โ ๐ง1๐ง๐ผ3 )(1โ ๐ง2๐ง๐ฝ3 )
=1
(๐ง๐ผ2 โ ๐ง๐ฝ1 )
(๐ (๐ง3)
(1โ ๐ง1๐ง๐ผ3 )+
๏ฟฝ๏ฟฝ(๐ง3)
(1โ ๐ง2๐ง๐ฝ3 )
)(3.8)
3.1. PARTITION ANALYSIS REVISITED 79
where
๐ (๐ง3) :=๐ผโ1โ๐=0
๏ฟฝ๏ฟฝ๐๐ง๐3 for ๐๐ =
โงโจโฉโ๐ง๐ฝ1 ๐ง๐๐ฝ
2 if ๐ฝ|๐ or ๐ = 0,
โ๐งrmd((๐ผโ1 mod ๐ฝ)๐,๐ฝ)1 ๐ง
rmd((๐ฝโ1 mod ๐ผ)๐,๐ผ)2 otherwise,
while
๏ฟฝ๏ฟฝ(๐ง3) :=
๐ฝโ1โ๐=0
๏ฟฝ๏ฟฝ๐๐ง๐3 for ๏ฟฝ๏ฟฝ๐ =
{๐ง๐ผ2 if ๐ = 0,
๐งrmd((๐ผโ1 mod ๐ฝ)๐,๐ฝ)1 ๐ง
rmd((๐ฝโ1 mod ๐ผ)๐,๐ผ)2 otherwise.
๏ฟฝ
Moreover, two new base cases are introduced. As before, we define some notation forhomogeneous polynomials.
Definition 3.3 (Oblique Complete Homogeneous Polynomials, see [5])For ๐1, ๐2, . . . , ๐๐ โ N*, we define โ๐(๐ง1, ๐ง2, . . . , ๐ง๐; ๐1, ๐2, . . . , ๐๐) through the generatingfunction
โโ๐=0
โ๐(๐ง1, ๐ง2, . . . , ๐ง๐; ๐1, ๐2, . . . , ๐๐)๐ก๐ =
1
(1โ ๐ง1๐ก๐1)(1โ ๐ง2๐ก๐2) ยท ยท ยท (1โ ๐ง๐๐ก๐๐).
๏ฟฝ
The related partial sums are defined for ๐ โ Z, as
๐ป๐(๐ง1, ๐ง2, . . . , ๐ง๐; ๐1, ๐2, . . . , ๐๐) =
{โ๐๐=0 โ๐(๐ง1, ๐ง2, . . . , ๐ง๐; ๐1, ๐2, . . . , ๐๐) if ๐ โฅ 0
0 if ๐ < 0.
Lemma 9 (Case ๐ = 0, Section 2 in [5]). For any integer ๐,
ฮฉโฅ๐๐
(1โ ๐ฅ1๐๐1)(1โ ๐ฅ2๐๐2) . . . (1โ ๐ฅ๐๐๐๐)= ฮฉโฅ
โโ๐=0
โ๐(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐; ๐1, ๐2, . . . , ๐๐)๐๐+๐
=1
(1โ ๐ฅ1)(1โ ๐ฅ2) . . . (1โ ๐ฅ๐)
โ๐ปโ๐โ1(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐; ๐1, ๐2, . . . , ๐๐).
๏ฟฝ
Lemma 10 (Case ๐ = 0, Section 2 in [5]). For any integer ๐,
ฮฉโฅ๐๐
(1โ ๐ฆ1๐๐1
)(1โ ๐ฆ2๐๐2
) . . . (1โ ๐ฆ๐๐๐๐ )
= ฮฉโฅ
โโ๐=0
โ๐(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐; ๐1, ๐2, . . . , ๐๐)๐๐โ๐
= ๐ป๐(๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐; ๐1, ๐2, . . . , ๐๐).
๏ฟฝ
80 CHAPTER 3. PARTITION ANALYSIS
Lemma 11 (Case ๐ = 1, Section 2 in [5]). For any integer ๐ < ๐,
ฮฉโฅ๐๐
(1โ ๐ฅ1๐๐1) . . . (1โ ๐ฅ๐๐๐๐)(1โ ๐ฆ๐โ๐)=
1
(1โ ๐ฅ1)(1โ ๐ฅ2) . . . (1โ ๐ฅ๐)(1โ ๐ฆ)
โโ๐โ1
๐1,๐2,...,๐๐=0
โ๐ฅ๐๐๐ ๐ฆ
โโ๐๐๐๐+๐
๐
โ+1
(1โ ๐ฅ๐1๐ฆ๐1) . . . (1โ ๐ฅ๐๐๐ฆ
๐๐)(1โ ๐ฆ).
๏ฟฝ
Lemma 12 (Case ๐ = 1, Section 2 in [5]). For any integer ๐ > โ๐,
ฮฉโฅ๐๐
(1โ ๐ฅ๐๐)(1โ ๐ฆ1๐โ๐1) . . . (1โ ๐ฆ๐๐โ๐๐)=
โ๐โ1๐1,๐2,...,๐๐=0
โ๐ฆ๐๐๐ ๐ฅ
โโ๐๐๐๐โ๐
๐
โ(1โ ๐ฅ)(1โ ๐ฅ๐1๐ฆ๐1) . . . (1โ ๐ฅ๐๐๐ฆ๐๐)
.
๏ฟฝ
Omega2 was used to obtain results in a series of papers by Andrews-Paule andtheir coauthors, dealing with various problems from partition theory. For the project ingeneral see http://www.risc.jku.at/research/combinat/software/Omega/.
Now we proceed with the geometric interpretation of the Generalized Partial FractionDecomposition. We first rewrite (3.8) by pulling out โ๐ง๐ฝ1 from the denominator of
1
(๐ง๐ผ2 โ๐ง๐ฝ1 ),i.e.,
1
(1โ ๐ง1๐ง๐ผ3 )(1โ ๐ง2๐ง๐ฝ3 )
=โ๐งโ๐ฝ
1 ๐ (๐ง3)
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง1๐ง๐ผ3 )
โ ๐งโ๐ฝ1 ๏ฟฝ๏ฟฝ(๐ง3)
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง2๐ง
๐ฝ3 )
, (3.9)
and define ๐๐ผ,๐ฝ and ๐๐ผ,๐ฝ to be
โ ๐๐ผ,๐ฝ := โ๐งโ๐ฝ1 ๐ (๐ง3) =
โ๐ผโ1๐=0 ๐๐๐ง
๐3
โ ๐๐ผ,๐ฝ := ๐งโ๐ฝ1 ๏ฟฝ๏ฟฝ(๐ง3) =
โ๐ฝโ1๐=0 ๐๐๐ง
๐3
where
โ ๐๐ =
โงโจโฉ๐ง๐๐ฝ
2 if ๐ฝ|๐ or ๐ = 0,
๐งrmd((๐ผโ1 mod ๐ฝ)๐,๐ฝ)โ๐ฝ1 ๐ง
rmd((๐ฝโ1 mod ๐ผ)๐,๐ผ)2 otherwise,
โ ๐๐ =
{๐งโ๐ฝ1 ๐ง๐ผ2 if ๐ = 0,
๐งrmd((๐ผโ1 mod ๐ฝ)๐,๐ฝ)โ๐ฝ1 ๐ง
rmd((๐ฝโ1 mod ๐ผ)๐,๐ผ)2 otherwise.
The following theorem provides the basis for the geometric interpretation of theGeneralized Partial Fraction Decomposition.
3.1. PARTITION ANALYSIS REVISITED 81
Theorem 3.2 (Polyhedral geometry interpretation of Omega2)Application of the generalized partial fraction decomposition
1
(1โ ๐ง1๐ง๐ผ3 )(1โ ๐ง2๐ง๐ฝ3 )
=๐๐ผ,๐ฝ
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง1๐ง๐ผ3 )
โ๐๐ผ,๐ฝ
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง2๐ง
๐ฝ3 )
(3.10)
on ๐๐ถ(z) induces a signed cone decomposition 1 of the cone ๐ถ = ๐R ((1, 0, ๐ผ), (0, 1, ๐ฝ)).๏ฟฝ
Proof Strategy:
โ Determine the structure of the fundamental parallelepiped of the cones ๐ด =๐R ((โ๐ฝ, ๐ผ, 0), (1, 0, ๐ผ)) and ๐ต = ๐R ((โ๐ฝ, ๐ผ, 0), (0, 1, ๐ฝ)).
โ Prove that ๐๐ผ,๐ฝ and ๐๐ผ,๐ฝ are the generating functions of these fundamental par-allelepipeds.
For the detailed proof of Theorem 3.2 see Appendix A. Here we provide the pic-ture of the decomposition for the cone ๐ถ = ๐R ((1, 0, 3), (0, 1, 5)), into the cones ๐ด =
๐R ((โ5, 3, 0), (1, 0, 3)) and ๐ต = ๐0,1R ((โ5, 3, 0), (0, 1, 5)).
๐ฅ
๐ฆ
๐ง(1, 0, 3)
(0, 1, 5) ๐ฅ
๐ฆ
๐ง
(โ5, 3, 0)
(1, 0, 3)(0, 1, 5)
1A decomposition into a sum where the summands may have positive or negative sign.
82 CHAPTER 3. PARTITION ANALYSIS
๐ฅ
๐ฆ
๐ง(1, 0, 3)
(โ5, 3, 0)
(0, 1, 5)๐ฅ
๐ฆ
๐ง(1, 0, 3)
(โ5, 3, 0)
(0, 1, 5)
The statement (from the proof in Appendix A)
๐ฮ (๐ต) = ๐๐ผ,๐ฝ + 1โ ๐งโ๐ฝ1 ๐ง๐ผ2
means that๐๐ผ,๐ฝ
(1โ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ๐ง2๐ง
๐ฝ3 )
is the generating function of the half-open cone ๐ต that is
open on the ray generated by (โ๐ฝ, ๐ผ, 0).One can see the full signed decomposition on the generating-function level as follows:
๐๐ถ = ๐๐ด โ ๐๐ต + ๐๐R((0,1,๐ฝ)).
According to the previous analysis ๐๐ด =๐๐ผ,๐ฝ
(1โ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ๐ง1๐ง๐ผ3 )
.
Since ๐๐R((0,1,๐ฝ))= 1
(1โ๐ง2๐ง๐ฝ3 )
we have
โ๐๐ต + ๐๐R((0,1,๐ฝ))= โ
๐๐ผ,๐ฝ + 1โ ๐งโ๐ฝ1 ๐ง๐ผ2
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง2๐ง
๐ฝ3 )
+1(
1โ ๐ง2๐ง๐ฝ3
)= โ
๐๐ผ,๐ฝ
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง2๐ง
๐ฝ3 )
which means
1
(1โ ๐ง1๐ง๐ผ3 )(1โ ๐ง2๐ง๐ฝ3 )
=๐๐ผ,๐ฝ
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง1๐ง๐ผ3 )
โ๐๐ผ,๐ฝ
(1โ ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ ๐ง2๐ง
๐ฝ3 )
.
In other words, ๐๐ผ,๐ฝ encodes the inclusion-exclusion step in the decomposition.
3.2. ALGEBRAIC PARTITION ANALYSIS 83
3.2 Algebraic Partition Analysis
Having reviewed the history of partition analysis both from the perspective of the originalauthors and from a more geometric perspective, we give now an algebraic presentationthe ฮฉโฅ operator .
In particular, we will use graded vector spaces and rings in order to compute thegenerating functions of the solutions of a linear Diophantine system.
3.2.1 Graded Rings
Let ๐ = K [๐ง1, ๐ง2, . . . , ๐ง๐] be the ring of polynomials in ๐ variables ๐ง1, ๐ง2, . . . , ๐ง๐. In whatfollows we will grade this ring in a way that is useful for solving linear Diophantinesystems. Let us first define a graded vector space, as we will look at the polynomial ringas a vector space over the coefficient field.
Definition 3.4 (Graded vector space)Let K be a field and ๐ a vector space of K. If there exists a direct sum decomposition
๐ =โจ๐โ๐ผ
๐๐
of ๐ into linear subspaces ๐๐ for some index set ๐ผ, then we say that ๐ is ๐ผ-gradedand the ๐๐โs are called homogeneous components, while the elements of ๐๐ are calledhomogeneous elements of degree ๐. ๏ฟฝ
The most usual grading of the polynomial ring K [๐ง1, ๐ง2, . . . , ๐ง๐] is the one inducedby the usual polynomial degree. The homogeneous component ๐๐ is the linear span ofall monomials of degree ๐, i.e., the set of all homogeneous polynomials of degree ๐ andthe polynomial 0.
A notion that connects graded algebraic structures with generating functions is thatof the Hilbert-Poincare series. Given a graded vector space ๐ =
โจ๐โ๐ผ ๐๐ for some
(appropriate) index set ๐ผ, then the Hilbert-Poincare series of ๐ over K is the formalpowerseries
โ๐ซ (๐ ) =โ๐โ๐ผ
dimK (๐๐) ๐ก๐.
Note that:
โ The most usual definition of Hilbert-Poincare series in the literature, restricts theindex set ๐ผ to be N and all linear subspaces ๐๐ to be finite-dimensional.
โ If the index set ๐ผ is not a subset of N๐ for some ๐, but all elements of ๐ผ are boundedfrom below, then the Hilbert-Poincare series is a formal Laurent series. If the indexset contains elements that are not bounded from below, then the Hilbert-Poincareseries is not defined.
โ According to the definition of a graded vector space, it is possible that not all ๐๐
are finite-dimensional. In that case the Hilbert-Poincare series is not defined.
84 CHAPTER 3. PARTITION ANALYSIS
The algebraic approach to partition analysis has its roots in the very early work andmotivation of Cayley, MacMahon and Elliott. Their motivation to deal with the problemstem from invariant theory, which was very fashionable these days. We will present herea setup of algebraic structures and associated series, mostly Hilbert series, which allowsto view partition analysis algebraically. In the definition of graded vector spaces, andas is customary in literature, we first defined the homogeneous components and thenassign to the elements of each component the corresponding degree. For our purposesthough, it is more natural to follow the reverse path. We will first define a degree onthe elements of the polynomial ring and then construct the homogeneous components asthe linear span of all elements of the same degree. In particular, we want to constructa grading of K [๐ง1, ๐ง2, . . . , ๐ง๐] starting from a matrix ๐ด in Z๐ร๐. We assume that ๐ด isfull-rank.
Let ๐๐๐ be the linear functionals defined by the rows of ๐ด for ๐ โ [๐]. We define thefunction ๐น : N๐ โฆโ Z๐ by
๐น (๐ฃ) = (๐๐1(๐ฃ), ๐๐2(๐ฃ), . . . , ๐๐๐(๐ฃ)) for ๐ฃ โ N๐.
Since we want to use this function to grade the polynomial ring, it is necessary totranslate from vectors to monomials and back. Given a set of formal variables ๐ ={๐ง1, ๐ง2, . . . , ๐ง๐}, we define the term monoid T to be the subset of K [๐ง1, ๐ง2, . . . , ๐ง๐] con-sisting of powerproducts, i.e., {๐ง๐11 ๐ง๐22 ยท ยท ยท ๐ง
๐๐๐ |๐๐ โ N} and we set 1 = ๐ง0. This is a
multiplicative monoid with the standard monomial multiplication. We define ๐ as
๐ :N๐ โ T๐ฃ โฆโ ๐ง๐ฃ
establishing the monoid isomorphism N๐ โ T. Now, we can define a degree in T via ๐น .
N๐
T Z๐
โ ๐น
deg
In words, let deg : Tโ Z๐ be defined as deg(๐ง๐ฃ) = ๐น(๐โ1(๐ง๐ฃ)
).
Define ๐๐ for ๐ โ Z๐ to be the linear span over K of all elements in T with degree ๐, i.e.,
๐๐ =
{๐โ
๐=0
๐๐๐ก๐
๐ โ N, ๐๐ โ K, ๐ก๐ โ T with deg ๐ก๐ = ๐
}for ๐ โ Z๐
= โจ๐ก โ T : deg(๐ก) = ๐โฉK, for ๐ โ Z๐
and โจโ โฉK = {0}. Then we have that
โ = โ๐โZ๐๐๐.
This means that the collection of ๐๐โs defines a grading on K [๐ง1, ๐ง2, . . . , ๐ง๐], which nowcan be seen as a graded vector space. Let us see an example of such a grading.
3.2. ALGEBRAIC PARTITION ANALYSIS 85
Example 9 (Total degree or โ1-grading). The degree function is given by ๐น (๐ ) = |๐ |1,i.e., the matrix ๐ด is the unit matrix of rank ๐. This is the usual total degree for polyno-mials in ๐ variables. In the figure we see the grading for ๐ = 2.
10
9
8
7
6
0 1 2 3 4 5
๐๐ is the vector space of homogeneous polynomials of degree ๐.
๏ฟฝ
Example 10. Let the degree map be defined by ๐น (๐ ) = ๐ . The grading this map inducesis called trivial because each K-vector space ๐๐ is generated by ๐ง๐ alone.
๏ฟฝ
We note that some (or all) ๐๐ may be infinite dimensional depending on the matrix ๐ดwe started with, as in the following example.
Example 11. Let the degree map be defined by ๐น (๐ฅ, ๐ฆ) = ๐ฅ โ ๐ฆ. The grading thismap induces is infinite, because each K-vector space ๐๐ is generated by infinitely manymonomials.
86 CHAPTER 3. PARTITION ANALYSIS
โ1โ2โ3โ4โ5 0
1
2
3
4
5
๏ฟฝ
The Hilbert-Poincare series provide a useful, refined counting tool for the numberof solutions of linear Diophantine systems. Nevertheless, we would like to allow fornon finite-dimensional graded components as well as for a tool enumerating rather thancounting solutions.
Each ๐๐ is a vector space containing homogeneous polynomials of degree ๐. It is alwayspossible to pick a basis of ๐๐ consisting of terms of degree ๐, i.e., elements of T. We willdenote such a basis by โฌ๐. We define the basis of {0} to be the empty set.
In the direction of obtaining an enumerating generating function we define the trun-cated multivariate Hilbert-Poincare series,
Definition 3.5 (truncated multivariate Hilbert-Poincare series)Given a graded vector space ๐ =
โจ๐โZ๐ ๐๐ over the field K and a vector ๐ โ Z๐ for some
๐ โ N, we define the truncated multivariate Hilbert-Poincare series of ๐ as
๐กโ๐ซ๐๐ (๐ง, ๐ก) =
โ๐โค๐โZ๐
โโโ๐โโฌ๐
๐
โโ ๐ก๐.
The truncated multivariate Hilbert-Poincare series is a formal Laurent series in the ๐กvariables and a formal powerseries in the ๐ง variables. ๏ฟฝ
The truncation in the indices is necessary in order to ensure that the formal Laurentseries is well defined, while the formal sum of the basis elements avoid the problem ofinfinite-dimensionality. At the same time, this formal series enumerates all solutions ofa linear Diophantine system ๐ด๐ฅ โฅ ๐, if the vector space ๐ is graded via the proceduredescribed above for the matrix ๐ด โ Z๐ร๐. We note that in that case ๐ = ๐.
The following theorem provides the connection between the ฮฉโฅ operator and Hilbert-Poincare series.
3.2. ALGEBRAIC PARTITION ANALYSIS 87
Theorem 1. Given ๐ด โ Z๐ร๐, ๐ โ Z๐ and ๐กโ๐ซ๐๐ (๐ง, ๐ก) as above, we have
ฮฉโฅโ๐ฅโN๐
๐ง๐ฅ๐โ๐=1
๐๐ด๐๐ฅโ๐๐๐ = ๐กโ๐ซ๐
๐ (๐ง, ๐ก)|๐ก=1.
๏ฟฝ
Proof. Let ๐ be the set of solutions to the linear Diophantine system ๐ด๐ฅ โฅ ๐. By thedefinition of the ฮฉโฅ operator, we have that ฮฉโฅ
โxโN๐ zx
โ๐๐=1 ๐
๐ด๐xโ๐๐๐ is ฮฆ๐ด,๐(๐ง), the
generating function of ๐. Now we have:
๐ผ โ ๐ โ ๐ด๐ผ โฅ ๐
โ ๐ด๐ผ = ๐ with ๐ โฅ ๐
โ ๐น (๐ผ) = ๐ with ๐ โฅ ๐
โ deg(๐ง๐ผ) = ๐ with ๐ โฅ ๐
โ ๐ง๐ผ โ ๐ ๐ with ๐ โฅ ๐
โ ๐ง๐ผ โ ๐ต๐ with ๐ โฅ ๐.
The last equivalence implies that
[๐ง๐ฅ]ฮฆ๐ด,๐(๐ง) = 1โ [๐ง๐ฅ]๐กโ๐ซ๐๐ (๐ง, ๐ก) = ๐ก๐น (๐ฅ)
and the theorem follows.
Note that this relation is all but new. It is actually as old as Cayley who first used theฮฉโฅ operator in the context of invariant theory.
The construction of the grading provides a way to describe the solutions of a linearDiophantine system. Given a matrix ๐ด in Z๐ร๐, the solution set of the homogeneouslinear Diophantine system ๐ด๐ฅ โฅ 0 isโ
๐โN๐
๐โ1 (โฌ๐) โ N๐.
For the inhomogeneous linear Diophantine system ๐ด๐ฅ โฅ ๐ for some ๐ โ Z๐ we have thatthe solution set is โ
๐โค๐โN๐
๐โ1 (โฌ๐) โ N๐.
In other words, the solutions of a linear Diophantine system or equivalently the latticepoints of a polyhedron can be described via the construction of the appropriate gradinggiven by the inequality description.
We will now proceed with a different grading construction, relevant to polytopes andEhrhart theory.
88 CHAPTER 3. PARTITION ANALYSIS
Ehrhart grading
As we saw, the description of the lattice points in a polytope can be given by the aboveprocedure. Continuing in the same direction, we will construct a grading that instead ofjust describing the lattice points in a polytope, it gives the lattice points in a cone overa polytope. The cone over a polytope is a fundamental construction in Ehrhart theory,used to enumerate the lattice points in the dilations of a polytope. Given a polytope ๐ in R๐, we embed it in R๐+1 at height 1, i.e., ๐ โฒ =
{(๐ฅ, 1) โ R๐+1
๐ฅ โ ๐
}.
In our grading construction, we respect the decomposition of lattice points according totheir height. Given a matrix ๐ด in Z๐ร๐ and a vector ๐ in Z๐ we construct the matrix
๐ธ =
โกโขโขโขโขโขโขโฃโ๐1
๐ด...
โ๐๐
0 . . . 0 1
โคโฅโฅโฅโฅโฅโฅโฆ
This matrix has the property that if the matrix ๐ด defines a polytope, then the first ๐rows of matrix ๐ธ define the ๐ก-dilate of that polytope. The last row of ๐ธ is used to keeptrack of the dilate ๐ก.
Since we allow the polytope ๐ to contain negative coordinates, we consider the Laurentpolynomial ring in ๐+1 variables K
[๐งยฑ11 , ๐งยฑ1
2 , . . . , ๐งยฑ1๐ , ๐กยฑ1
]and the corresponding Lau-
rent term monoid T = [๐งยฑ11 , ๐งยฑ1
2 , . . . , ๐งยฑ1๐ , ๐กยฑ1]. We construct the ๐๐ as before using the
matrix ๐ธ.
Now, on top of the set of ๐๐ we need some extra structure. For ๐ in Z, define ๐ด๐ to bethe direct sum of all ๐๐ for which the last coordinate of ๐ is equal to ๐, i.e.,
๐ด๐ =โจโจ๐ก โ T | deg(๐ก) = (๐ผ1, ๐ผ2, . . . , ๐ผ๐, ๐)โฉK
The vector space ๐ is now graded by the collection of ๐ด๐โs. One should not confuse thedeg(๐ฃ) of an element ๐ฃ of ๐ , which is a vector in Z๐+1 and the degree of ๐ฃ induced bythe grading, which is a non-negative integer indicating the height (last coordinate) of๐โ1(๐ฃ) in the cone over the polytope.
3.2. ALGEBRAIC PARTITION ANALYSIS 89
5
4
3
2
1
0
โ1
โ2
โ3
โ4
โ5
Although this grading respects the height given by the cone over the polytope con-struction, it gives no information about which lattice points lie in the polytope. Notethat we are only interested in non-negative dilations. For this reason, we define thesubspaces ๐ฟ๐ as
๐ฟ๐ = โจ๐ก โ T | deg(๐ก) = (๐ผ1, ๐ผ2, . . . , ๐ผ๐, ๐) , ๐ผ๐ โฅ 0 for all ๐ โ [๐]โฉK
for ๐ โ N, i.e., each ๐ฟ๐ is generated by the monomials of ๐ด๐ that correspond to latticepoints in the ๐-th dilation of the polytope. This is expressed by the non-negativitycondition on the first ๐ coordinates of the degree vector.
Contrary to the general construction from an arbitrary matrix ๐ด โ Z๐ร๐, the con-struction of a cone over a polytope guarantees that the homogeneous components ๐ฟ๐
are finite-dimensional subspaces. This is because at each height ๐, the ๐ฟ๐ is generatedby the finitely many lattice points in th ๐-th dilation of the (by definition bounded)polytope ๐ .
Now, the Ehrhart series of ๐ is given byโ
๐โN ๐๐๐K (๐ฟ๐) ๐ก๐.
Oblique Graded Simplices
A useful example for truncated multivariate Hilbert-Poincare series is the case of ObliqueGraded Simplices. Letโs see two cases:
โ If we use the grading induced by the total degree (โ1 grading) on the positivequadrant we have the following picture
90 CHAPTER 3. PARTITION ANALYSIS
and the corresponding truncated multivariate Hilbert-Poincare series is the gener-ating function for the lattice points in the positive quadrant graded by the givengrading. Note that if ๐ is the grading constructed as above, then
โโฒ๐(๐ง) = [๐ก๐]๐กโ๐ซ๐ (๐ง, ๐ก)
are the usual complete homogeneous symmetric polynomial of degree ๐.
โ If we use the degree function given by degree(๐ฅ, ๐ฆ) = ๐ฅ+ 2๐ฆ the picture is
and thenโโฒ๐(๐ง) = [๐ก๐]๐กโ๐ซ๐ (๐ง, ๐ก)
are the oblique complete homogeneous polynomials โ๐(๐ง; 1, 2).
In [5], the authors introduce โ๐(๐ง1, ๐ง2, . . . , ๐ง๐; ๐1, ๐2, . . . , ๐๐) for ๐๐ โ N, โa variant ofthe symmetric functionsโ, given by their generating function
โโ๐=0
โ๐(๐ง1, ๐ง2, . . . , ๐ง๐; ๐1, ๐2, . . . , ๐๐)๐ก๐ =
1
(1โ ๐ง1๐ก๐1)(1โ ๐ง2๐ก๐2) ยท ยท ยท (1โ ๐ง๐๐ก๐๐).
Chapter 4
Partition Analysis via PolyhedralGeometry
The main idea behind all implementations of the ฮฉโฅ operator is to obtain a partial-fraction decomposition of the crude generating function and exploit the linearity of theฮฉโฅ operator. In particular, we want a partial-fraction decomposition such that in thedenominator of each fraction appear either only non-negative or only non-positive powersof ๐โs. Given such a partial-fraction decomposition, we can be sure that in the seriesexpansions of the crude generating function with respect to the ๐โs, the fractions withonly non-positive powers of ๐ cannot contribute terms with positive ๐ exponents. Inother words, we should only keep the fractions that have non-negative powers of ๐ andthe fractions that are ๐-free. There are different ways to compute such a partial-fractiondecomposition, like Elliottโs algorithm [24] and Omega2 [5].
In this chapter, we present an algorithmic geometric approach. The main goal ofthe method presented here is to compute a cone decomposition of the ฮฉ-polyhedron.Instead of going directly for the rational generating function of the lattice points in theฮฉ-polyhedron, we first compute a set of simplicial cones that sum up to the desiredpolyhedron. This step is done by using only rational linear algebra. In order to obtainthe rational generating function, one can either use Barvinokโs algorithm or explicitformulas. Some more custom-tailored tools will appear in [19]. Here we restrict tocomputing a set of โsymbolic conesโ, i.e., instead of using the actual generating functionwe say โthe cone generated by ๐1, ๐2, . . . , ๐๐ with apex ๐โ. This work is part of [19].
Recall the definitions of the rational form of the crude generating function
๐ฮฉ๐ด,๐(๐ง;๐) = ฮฉโฅ๐โ๐โ๐
๐=1(1โ ๐ง๐๐๐ด๐)
and of the formal Laurent series form
ฮฆฮฉ๐ด,๐(๐ง;๐) = ฮฉโฅ
โ๐ฅ1,...,๐ฅ๐โN
๐๐ด๐ฅโ๐๐ง๐ฅ11 . . . ๐ง๐ฅ๐
๐ ,
91
92 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
as well as that of the formal Laurent series form of the ๐-generating function
ฮฆ๐๐ด,๐(๐ง) =
โ๐ฅ1,...,๐ฅ๐โN
๐๐ด๐ฅโ๐๐ง๐ฅ11 . . . ๐ง๐ฅ๐
๐
and its rational form
๐๐๐ด,๐(๐ง) =๐โ๐โ๐
๐=1(1โ ๐ง๐๐๐ด๐).
We will denote by๐ด the matrix with๐ด1, ๐ด2, . . . , ๐ด๐ as columns. We note that ๐ฮ ๐ถ(๐ง) and
ฮฆฮ ๐ถ(๐ง), where ฮ ๐ถ denotes the fundamental parallelepiped of the cone ๐ถ, are identical
since the fundamental parallelepiped is finite and both objects collapse to a Laurentpolynomial.
4.1 Eliminating a single ๐
Elimination of a single ๐ amounts to the solution of a single linear Diophantine inequality.All known implementations of the ฮฉโฅ operator are based on the recursive eliminationof ๐โs for the solution of linear Diophantine systems. In order to employ polyhedralgeometry, we need to translate the problem to a problem about cones. Recall thedefinitions of the ๐-cone and the ฮฉ-polyhedron:
๐๐,Z (๐ด) = ๐R ((๐1 : ๐ด1), (๐2 : ๐ด2), . . . , (๐๐ : ๐ด๐))
๐ซฮฉโฅ,Rb (๐ด) = {x โ R๐
+ : ๐ดx โฅ b}Let ๐ป๐ = {(๐ฅ1, . . . , ๐ฅ๐, ๐ฅ๐+1) : ๐ โค ๐ฅ๐+1} be the set of vectors with last componentgreater or equal to ๐. Since we have one ๐, the ๐ด๐ is an integer denoted by ๐๐.
Computing ฮฉโฅ๐โ๐
(1โ๐ง1๐๐1 )ยท...ยท(1โ๐ง๐๐๐๐ ) amounts to computing the generating function
๐๐(๐ถโฉ๐ป๐)(๐ง1, . . . , ๐ง๐), where ๐ denotes projection with respect to the last coordinate, as
follows:
1. Compute a vertex description of the vertex cones ๐ฆ๐ฃ at the vertices of ๐ถ โฉ๐ป๐.
2. Compute the generating functions ๐๐ฆ๐ฃ(๐ง1, ๐ง2, . . . , ๐ง๐+1), either using Barvinokโs
algorithm [12] or by explicit formulas using modular arithmetic (similar to theexpressions in [5]).
3. Substitute ๐ง๐+1 โฆโ 1 to obtain ๐-free generating functions (projection of the poly-hedron).
4. Sum all the projected generating functions for the tangent cones. By Brionโstheorem [12], this yields the desired generating function ฮฉโฅ
๐โ๐
(1โ๐ง1๐๐1 )ยท...ยท(1โ๐ง๐๐๐๐ ) .
Let ๐ be the polyhedron ๐ถ โฉ๐ป๐. We need to compute the vertices and vertex conesof ๐ . We note that any polyhedron can be written as the Minkowski sum of a cone anda polytope. In our case
๐ = conv(0, ๐ข๐ : ๐ โ {1, . . . , โ}) + ๐R (๐ค๐,๐ : ๐ โ {1, . . . , โ}, ๐ โ {โ+ 1, . . . , ๐})
4.1. ELIMINATING ONE VARIABLE 93
where
๐ข๐ =๐
๐๐๐ฃ๐ = (. . .
๐
๐๐. . . ๐)
๐ค๐,๐ = โ๐๐๐ฃ๐ + ๐๐๐ฃ๐ = (0 . . . 0 โ๐๐ 0 . . . 0 ๐๐ 0 . . . 0).
The vertices of ๐ , denoted by ๐ข๐, are the intersection points of the hyperplane at height๐ฅ๐+1 = ๐ with the generators of ๐ that are pointing โupโ (๐๐ โฅ 0). The ๐ค๐,๐ are positivelinear combinations of the generators ๐ฃ๐ that are pointing โupโ and the generators ๐ฃ๐that are pointing โdownโ (๐๐ < 0) such that ๐ค๐,๐ has last coordinate equal to zero.Combinatorially, this is a cone over a product of simplices.
Next, we want to compute the tangent cones of our polyhedron ๐ . The tangentcones are then given by the following lemma.
Lemma 13. The generators of the cone ๐ฆ๐ข๐ are
โ๐ข๐ and ๐ข๐โฒ โ ๐ข๐ for ๐โฒ โ {1, . . . , โ} โ {๐} and ๐ค๐,๐ for ๐ โ {โ+ 1, . . . , ๐}.
These are ๐ generators in total that are linearly independent. In particular, ๐ฆ๐ข๐ is asimplicial cone.
๏ฟฝ
Proof. First, we argue that ๐ค๐โฒ,๐ is not a generator of ๐ฆ๐ข๐ for all ๐โฒ โ {1, . . . , โ} โ {๐} and
all ๐ โ {โ+ 1, . . . , ๐}. This follows from the simple calculation
๐ข๐ โ๐
๐๐โฒ๐๐๐ค๐โฒ,๐ =
โกโขโขโขโขโขโขโฃ๐๐๐
0
0
๐
โคโฅโฅโฅโฅโฅโฅโฆโ๐
๐๐โฒ๐๐
โกโขโขโขโขโขโขโฃ0
โ๐๐
๐๐โฒ
0
โคโฅโฅโฅโฅโฅโฅโฆ =
โกโขโขโขโขโขโขโฃ0
๐๐๐โฒ
0
๐
โคโฅโฅโฅโฅโฅโฅโฆโ๐
๐๐๐๐
โกโขโขโขโขโขโขโฃโ๐๐
0
๐๐
0
โคโฅโฅโฅโฅโฅโฅโฆ = ๐ข๐โฒ โ๐
๐๐๐๐๐ค๐,๐
where the four components of the vectors that are shown have indices ๐, ๐โฒ, ๐, ๐ + 1, inthat order, all other components are zero. Note that the coefficients โ ๐
๐๐โฒ๐๐and โ ๐
๐๐๐๐are positive rational numbers as ๐๐ is negative.Moreover, none of the vectors ๐ข๐โฒ โ ๐ข๐ for ๐โฒ = ๐ = ๐ are in the tangent cone. This canbe seen by observing that the ๐-th component of ๐ข๐โฒ โ ๐ข๐ is negative, while the ๐-thcomponent of ๐ข๐ is zero. However, ๐ โ R๐
โฅ0 ร R.So now we know that the only directions that can appear as generators are those given inthe lemma. That all of these are, in fact, generators follows from a dimension argument:There are precisely ๐ directions given in the lemma and we know that the polyhedron๐ and all of its tangent cones have dimension ๐ as well. So the tangent cones have tohave at least ๐ generators.
We denote by ๐๐ the matrix that has as columns the vectors ๐๐๐ข๐, lcm(๐๐, ๐๐โฒ)(๐ข๐โฒ โ ๐ข๐)for ๐โฒ โ {1, . . . , โ} โ {๐} and ๐ค๐,๐ for ๐ โ {โ+ 1, . . . , ๐}. ๐๐ is a (๐+ 1)ร ๐ integer matrixthat has a full-rank diagonal submatrix.
94 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
By Brionโs theorem, the generating function for ๐(๐โฉ๐ป๐) is the sum for all ๐ of the gener-ating function of the cone generated by the columns of ๐๐ and equal to ๐๐ถ(๐ง1, ๐ง2, . . . , ๐ง๐).We can now present the algorithm:
Algorithm 2 Elimination of a single ๐
Require: ๐ โ Z and ๐ด โ Z๐ such that ๐1, ๐2, . . . , ๐โ > ๐ and ๐โ+1, ๐โ+2, . . . , ๐๐ < ๐
1: ๐ผ+ โ {1, . . . , โ}2: ๐ผ+ โ {โ+ 1, . . . , ๐}3: for ๐ โ ๐ผ+ do4: ๐ฃ๐ โ (๐๐ : ๐๐)
5: ๐ โ {(. . . ๐๐๐. . . ๐)} for ๐ โ ๐ผ+
6: for ๐ โ ๐ผ+ do7: for ๐ โ ๐ผโ do8: ๐ค๐,๐ โ โ๐๐๐๐ + ๐๐๐๐
9: for ๐ โ ๐ผ+ do10: ๐๐ โ {โ๐ข๐, ๐ข๐โฒ โ ๐ข๐, ๐ค๐,๐ : ๐
โฒ โ ๐ผ+ โ {๐}, ๐ โ ๐ผโ}11: ๐๐ โ a triangulation of the vertex cone ๐R (๐๐)
12: ๐๐R(๐๐)(๐ง1, ๐ง2, . . . , ๐ง๐+1) โ
โ๐ โ๐๐
๐๐ (๐ง1, ๐ง2, . . . , ๐ง๐+1)
13: ๐(๐ง1, ๐ง2, . . . , ๐ง๐+1) โโ๐โ๐ผ+
๐๐๐ฃ(๐ง1, ๐ง2, . . . , ๐ง๐+1)
14: return ๐(๐ง1, ๐ง2, . . . , ๐ง๐, 1)
To illustrate the algorithm we present an example.
Example 12. Given an inequality 2๐ฅ1 + 3๐ฅ2 โ 5๐ฅ3 โฅ 4, we want to compute
ฮฉโฅ๐โ4
(1โ ๐ง1๐2)(1โ ๐ง2๐3)(1โ ๐ง3๐โ5).
We first compute the generators of ๐๐,Z (๐ด):
๐ฃ1 = (1, 0, 0, 2), ๐ฃ2 = (0, 1, 0, 3), ๐ฃ3 = (0, 0, 1,โ5)
and construct the matrix ๐ :
๐ =
โกโขโขโขโขโขโขโฃ1 0 0
0 1 0
0 0 1
2 3 โ5
โคโฅโฅโฅโฅโฅโฅโฆ .
We want to compute the generating function of the intersection of ๐๐,Z (๐ด) and ๐ป4 ={(๐ฅ1, ๐ฅ2, ๐ฅ3, ๐ฅ4) โ R4 : ๐ฅ4 โฅ 4}.
4.1. ELIMINATING ONE VARIABLE 95
The vectors needed to construct the tangent cones are:
๐ข1 =4
2๐ฃ1 = (2, 0, 0, 4)
๐ข2 =4
3๐ฃ2 = (0,
4
3, 0, 4)
๐ค1,3 = (โ5, 0, 2, 0)
๐ค2,3 = (0,โ5, 3, 0)
and the two tangent cones are:
๐พ๐ข1 = ๐R((โ2, 0, 0,โ4), (โ2, 4
3, 0, 0), (โ5, 0, 2, 0); (2, 0, 0, 4)
)
๐พ๐ข2 = ๐R((0,โ4
3, 0,โ4), (2,โ4
3, 0, 0), (0,โ5, 3, 0); (0, 4
3, 0, 4)
).
The generating functions of the lattice points in the projected cones generated by thecolumns of ๐1 and ๐2 are
๐ง41(๐ง1 + ๐ง2 + ๐ง41๐ง3 + ๐ง21๐ง2๐ง3
)(1โ ๐ง1)
(๐ง31 โ ๐ง22
) (1โ ๐ง51๐ง
23
)and
โ๐ง22(๐ง21 + ๐ง1๐ง2 + ๐ง22 + ๐ง21๐ง
22๐ง3 + ๐ง32๐ง3 + ๐ง1๐ง
32๐ง3 + ๐ง1๐ง
42๐ง
23 + ๐ง21๐ง
42๐ง
23 + ๐ง52๐ง
23
)(1โ ๐ง2)
(๐ง31 โ ๐ง22
) (1โ ๐ง52๐ง
33
) .
By Brion their sum is the generating function for ๐(๐ถ โฉ๐ป4), which is equal to
ฮฉโฅ๐โ4
(1โ ๐ง1๐2)(1โ ๐ง2๐3)(1โ ๐ง3๐โ5).
๏ฟฝ
In the following figure we can see the ฮฉ-polyhedron and the two vertex cones ๐พ๐ข1
and ๐พ๐ข2 . Note that in the drawing our vectors are supposed to live in R4.
96 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
The two vertex cones. Elimination of a single ๐.
The discussion above and Lemma 13 give us the following theorem that summarizesthe algorithm for the elimination of one ๐. Note that instead of taking the halfspace๐ป๐, we translate the cone by โ๐ and use the positive halfspace with respect to the ๐coordinate denoted by ๐ป๐. The subscript does not mean the offset but denotes thecoordinate.
Theorem 2. Let ๐1, . . . , ๐๐ โ Q๐ be linearly independent vectors, let ๐ด โ Q๐ร๐ denotethe matrix with the ๐๐ as columns, let ๐ โ Q๐, and let ๐ผ โ [๐]. Then ๐๐ผR (๐ด; ๐) isan inhomogeneous simplicial cone. Assume that the affine hull of ๐๐ผR (๐ด; ๐) contains
integer points. Let ๐ โ [๐] and define ๐ปโฅ๐ = {๐ฅ โ R๐ | ๐ฅ๐ โฅ 0} to be the non-negative
half-space with respect to the coordinate ๐. Let โ+, โโ, โ0 denote the sets of indices๐ โ [๐] with ๐๐๐ > 0, ๐๐๐ < 0 and ๐๐๐ = 0, respectively. Let ๐ denote the polyhedron๐ = ๐ปโฅ
๐ โฉ (๐๐ผR (๐ด; ๐)). Under these assumptions, the following hold.If ๐๐ < 0, then:
1. The vertices of ๐ are
๐ฃ๐ = ๐โ ๐๐๐๐๐
๐๐ for ๐ โ โ+,
2. For every ๐ โ โ+, the extreme rays of the tangent cone of ๐ at ๐ฃ๐ are generated bythe vectors ๐1, . . . , ๐๐ defined by
๐๐ = ๐๐
๐๐โฒ = ๐๐โฒ โ ๐๐ for all ๐โฒ โ โ+ โ {๐}
๐๐ =1
๐๐๐๐๐ โ
1
๐๐๐๐๐ for all ๐ โ โโ
๐๐ = ๐๐ for all ๐ โ โ0.
The vectors ๐๐ are linearly independent whence the tangent cone at ๐ฃ๐ is simplicial.Let ๐บ๐ denote the matrix with these vectors as columns.
4.1. ELIMINATING ONE VARIABLE 97
3. The half-open tangent cone of ๐ at ๐ฃ๐ is
๐ฆ๐ฃ๐ = ๐ฃ๐ + ๐๐ผโ๐R (๐บ๐; 0) .
In particular, the tangent cone is โopen in direction ๐๐ if and only if the cone westarted out with is open in direction ๐๐ โ for all ๐ = ๐. It is closed in direction ๐,because we intersect with a closed half-space.
4. For any vertex ๐ฃ๐, let ๐ฝ๐ denote the set of indices ๐ โ [๐] where ๐ โ ๐ฝ๐ if and onlyif the first non-zero component of ๐๐ is negative (i.e. if ๐๐ is backward). Let ๐บโฒ
๐
denote the matrix obtained from ๐บ๐ by multiplying all columns with an index in ๐ฝ๐by โ1. (Now all columns of ๐บ๐ are forward.) Then
ฮฆ(๐๐ผ
R(๐ด;๐))โฉ๐ปโฅ๐
=โ๐โโ+
(โ1)|๐ฝ๐|ฮฆ๐R()๐ผโ๐ฮ ๐ฝ๐ (๐บโฒ
๐)+๐ฃ๐
where ฮ denotes the symmetric difference. The same identity also holds on thelevel of rational functions.
5. Let ๐ : R๐ โ R๐โ1 denote the projection that forgets the ๐-th coordinate. Ifaff(๐ถ) โฉ ๐ฟ = 0 for some linear subspace ๐ฟ that contains ker(๐), then
ฮฉ๐ฮฆ๐๐ผR(๐ด;๐)
=โ๐โโ+
(โ1)|๐ฝ๐|ฮฆ๐๐ผโ๐ฮ ๐ฝ๐R (๏ฟฝ๏ฟฝ๐;๐ฃ๐)
.
where ๏ฟฝ๏ฟฝ๐ denotes the matrix obtained from ๐บโฒ๐ by deleting the ๐th row. Moreover,
all the cones ๐ถ๐ = ๐๐ผโ๐ฮ ๐ฝ๐R
(๏ฟฝ๏ฟฝ๐; ๐ฃ๐
)have the property that aff(๐ถ๐) โฉ ๐(๐ฟ) = 0.
๏ฟฝ
We note that for the case when ๐๐ > 0, the theorem is completely analogous and theonly difference is that the apex of the cone is a vertex of the polyhedron. Each othervertex is visible from the apex, thus an extra ray is added in all vertex cones (from eachvertex to the apex of the cone). If ๐๐ = 0, then not all vertex cones are simplicial. Thuswe employ a deformation argument, presented by Fu Liu in [29] , and reduce the caseto the case ๐๐ < 0.
The following lemma says that after we project we can repeat the same procedure,thus allowing for recursive elimination of ๐s in order to solve systems of inequalities.
Lemma 14. Assume ker(๐) โ ๐ฟ. If ๐ฝ โฉ ๐ฟ = 0, then ๐(๐ฝ) โฉ ๐(๐ฟ) = 0. ๏ฟฝ
Proof. Let ๐ฃ โ ๐(๐ฝ)โฉ ๐(๐). Then there exist ๐ โ ๐ฝ and ๐ โ ๐ฟ such that ๐ฃ = ๐(๐) = ๐(๐).Because ๐ is linear, we have that ๐ โ ๐ โ ker(๐) โ ๐ฟ. Therefore ๐ = ๐ โ ๐+ ๐ โ ๐ฟ and so๐ โ ๐ฝ โฉ ๐ฟ which means by assumption ๐ = 0. But then ๐ฃ = ๐(๐) = 0.
98 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
4.2 Eliminating multiple ๐
Using the algorithm for single ๐ elimination recursively (due to Lemma 14, we essentiallycompute triangulations of the dual vertex cones and their generating functions. Thisis an interesting fact which comes directly from the particular choice of geometry thatpartition analysis makes. Nevertheless, recursive elimination may not always be themost desirable strategy, as it is well known that the choice of the elimination ordermay considerably alter the running time in the traditional partition analysis algorithms(intermediate expressions swell).
In order to eliminate multiple ๐ we need to follow the same procedure, but now thedimension of the ๐-cone is more than one lower than the ambient space, since we havemore than one ๐. This makes the computation of explicit expressions for the verticesand the vertex cones considerably harder. In what follows we give a description of howthis could be done algorithmically.
Given ๐ด โ Z๐ร๐ and ๐ โ Z๐ for ๐ โฅ 1, we define the vectors ๐ฃ๐ = (๐๐ : ๐ด๐) โ R๐+๐
where ๐ด๐ denotes the ๐-th column of ๐ด and ๐๐ the ๐-th standard unit vector in R๐. Let ๐denote the matrix with the ๐ฃ๐ as columns. Then ๐ is a vertex description of the ๐-cone
๐ถ = ๐๐,ZR (๐ด) = ๐R (๐ ) .
The ฮฉ-polyhedron we want to compute is now
๐ = ๐R (๐ ) โฉ{(๐ง, ๐) โ R๐+๐
๐๐ โฅ ๐๐ for all ๐ = 1, . . . ,๐
}.
Note that this definition of ๐ is neither a vertex-description nor a hyperplane-description.We need to compute the vertices and the tangent cones at the vertices of our polyhedronand we will follow a standard method for that [38].It is straightforward to give a hyperplane-description of ๐ถ. Note that ๐ถ is a simplicial๐-dimensional cone. Its facets are spanned by all combinations of ๐โ1 generators of thecone.
Lemma 15.
span(๐ถ) =
โงโจโฉโโ๐ฅ
โ
โโ โ R๐+๐
(๐ด โ๐ผ
)โโ๐ฅ
โ
โโ = 0
โซโฌโญ .
๏ฟฝ
Proof. There exist ๐ผ๐ such that ๐ฅ =โ
๐ ๐ผ๐๐ฃ๐ if and only if there exist ๐ผ๐ such that๐ฅ๐ = ๐ผ๐ for ๐ = 1, . . . , ๐ and โ๐ =
โ๐ ๐ผ๐๐๐,๐ for ๐ = 1, . . . ,๐. Such ๐ผ๐ exist if and only if
โ๐ =โ๐
๐=1 ๐ฅ๐๐๐,๐ for all ๐ = 1, . . . ,๐.
Given the linear span of ๐ถ, we can write down a hyperplane description of ๐ .
Lemma 16.
๐ =
โงโจโฉโโ๐ฅ
โ
โโ โ R๐+๐
โโ๐ฅ
โ
โโ โฅโโ0
๐
โโ and(๐ด โ๐ผ
)โโ๐ฅ
โ
โโ = 0
โซโฌโญ .
4.2. ELIMINATING MULTIPLE VARIABLES 99
๏ฟฝ
Proof. From the proof of the previous lemma, we see that ๐ฅ =โ
๐ ๐ผ๐๐ฃ๐ for ๐ผ๐ โฅ 0 if andonly if ๐ฅ๐ โฅ 0 for ๐ = 1, . . . , ๐ and
(๐ด โ๐ผ
)โโ๐ฅ
โ
โโ = 0.
So
๐ถ =
โงโจโฉโโ๐ฅ
โ
โโ โ R๐+๐
๐ฅ โฅ 0 and
(๐ด โ๐ผ
)โโ๐ฅ
โ
โโ = 0
โซโฌโญwhence the theorem follows from the definition of ๐ .
Having the H-description of the ฮฉ-polyhedron, what we need to do first is computeits vertices. This is done by considering all the full-rank square subsystems of the systemโโ๐ฅ
โ
โโ =
โโ0
๐
โโ (๐ด โ๐ผ
)โโ๐ฅ
โ
โโ = 0.
Each system has (at most) one rational solution. If the solution satisfies the full sys-tem (the point lies in the polyhedron), then that solution is a vertex of the ฮฉ-polyhedron.
Unfortunately, unlike the single ๐ case, explicit formulas are no more easy to obtainfor the vertex cones. This is due to the fact that there are many combinations ofpositive/negative entries in the inhomogeneous part. Thus, one can not simply labelgenerators as pointing up or down, since the same generator may be pointing up withrespect to one ๐ coordinate and down with respect to another.
We resort to the use of standard tools from polyhedral geometry.
โ Compute the vertex cones, which is possible since we have both an H-descriptionand a V-description of the polyhedron.
โ Triangulate if needed. The nice simplicial structure of the vertex cones is no longerguaranteed.
โ Use Brionโs theorem and Barvinokโs algorithm in order to compute the desiredrational generating function.
This method is algorithmic, but no closed formulas for the vertices and the vertexcones are provided. Nevertheless, while its performance will be poorer than that oftraditional recursive elimination algorithms in general, it is expected to perform betterthan traditional algorithms on problems that show intermediate expression swell. This
100 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
latter case is very often met in combinatorial problems, which have a nice 1 generatingfunction due to the structure of the problem, but the intermediate expressions do notsee the overall structure. One could say that this is a difference between local or globalknowledge about the problem structure.
1this usually means that a lot of cancellations occur in the rational function.
4.3. IMPROVEMENTS BASED ON GEOMETRY 101
4.3 Improvements based on geometry
4.3.1 Lattice Points in Fundamental Parallelepipeds
In this section we develop a closed formula for the generating function of the set of latticepoints in the fundamental parallelepiped of a simplicial cone. We will first illustrate thebasic idea for finding this lattice point set using an example.
Motivating Example
Let ๐ฝ โ R๐ be a lattice and ๐ฟ a sublattice with basis ๐1, . . . , ๐๐ โ ๐ฝ . Our goal is tocome up with a closed formula (or simply a way of enumerating) the ๐ฝ-lattice pointsin the fundamental parallelepiped generated by the ๐๐, i.e., we want to come up with agenerating function for the set ฮ (๐ด) where ๐ด is the ๐ร ๐-matrix with the vectors ๐๐ ascolumns.
As an example we will take ๐ฝ = Z2 and
๐ด =
โกโฃ 2 6
โ2 2
โคโฆ .
The fundamental parallelepiped ฮ (๐ด) and the lattice points contained therein are illus-trated in Figure 4.1.
Figure 4.1: The lattice points in the fundamental parallelepiped ฮ (๐ด).
If ฮ (๐ด) were a rectangle, or, more precisely, if ๐ด were a diagonal matrix, then๐ฝ โฉฮ (๐ด) would be particularly easy to describe. For example if ๐ด is an ๐ร ๐ diagonalmatrix with diagonal entries ๐1, . . . , ๐๐, then we have
ฮ (๐ด) = [0, ๐1)ร ยท ยท ยท ร [0, ๐๐)
Z๐ โฉฮ (๐ด) = {0, . . . , ๐1 โ 1} ร ยท ยท ยท ร {0, . . . , ๐๐ โ 1}
ฮฆฮ (๐ด)(๐ง) =1โ ๐ง๐111โ ๐ง1
ยท . . . ยท 1โ ๐ง๐๐๐1โ ๐ง๐
.
How can we make use of this simple observation about rectangles to handle generalmatrices ๐ด such as our example above? One idea is to transform ๐ด into a diagonal
102 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
matrix and control the corresponding changes to ฮ (๐ด). To implement this approach wecan use the Smith normal form of a matrix.
Theorem 4.1(Smith normal form) Let ๐ด be a non-zero ๐ร๐ integer matrix. Then there exist matrices๐ โ Z๐ร๐, ๐ โ Z๐ร ๐, ๐ โ Z๐ร๐ such that ๐ด = ๐๐๐ , the matrices ๐ and ๐ areinvertible over the integers (i.e., they are lattice transformations), and
๐ =
โกโขโขโขโขโขโขโขโขโขโขโขโขโขโขโขโขโฃ
๐ 1 0 0 ยท ยท ยท 0
0 ๐ 2 0 ยท ยท ยท 0
0 0. . .
...... ๐ ๐
0
. . .
0 0 0
โคโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฆ.
where ๐ is the rank of ๐ด and ๐ ๐|๐ ๐+1 and ๐ด = ๐๐๐ . ๏ฟฝ
In our case this can be interpreted as follows:
โ Intuitively, multiplying ๐ด from the left with a lattice transform ๐โ1 means per-forming elementary row operations on ๐ด, which corresponds to changing the basisof ๐ฝ . This simply applies a change of coordinates to the lattice points in thefundamental parallelepiped.
โ Multiplying ๐ด from the right with a lattice transform ๐ โ1 means performing el-ementary column operation on ๐ด, which corresponds to changing the basis of ๐ฟand thus changing the fundamental parallelepiped. We will return to the questionhow to relate ๐ฝ โฉฮ (๐ด) and ๐ฝ โฉฮ (๐ด๐ โ1) in a moment.
The result of applying both of these transformations to ๐ด is a diagonal matrix ๐. Sothe Smith normal form presents a method of changing bases for both ๐ฝ and ๐ฟ such thatthe fundamental parallelepiped is a rectangle with respect to the new bases.
Letโs see how this plays out in our example, see Figure 4.2. We start out with ๐1, ๐2as a basis for Z2 and ๐1, ๐2 as our basis for ๐ฟ. Our first step is to change bases on ๐ฟby replacing ๐1 with ๐โฒ1 = ๐1 โ ๐2 and keeping ๐โฒ2 = ๐2. We then have ๐1 = 2๐โฒ1. Thiscorresponds to the multiplication
๐โ1๐ด =
โกโฃ 1 0
1 1
โคโฆโกโฃ 2 6
โ2 2
โคโฆ =
โกโฃ 2 6
0 8
โคโฆ .
As we can see, the colums of the matrix on the right hand side give the coordinates of๐1, ๐2 in terms of ๐โฒ1, ๐
โฒ2.
4.3. IMPROVEMENTS BASED ON GEOMETRY 103
The next step is to changes bases of ๐ฟ and replace ๐2 with ๐โฒ2 = ๐2โ3๐1 and keeping๐โฒ1 = ๐1. This corresponds to the multiplication
(๐โ1๐ด)๐ โ1 =
โกโฃ 2 6
0 8
โคโฆ .
โกโฃ 1 โ3
0 1
โคโฆ =
โกโฃ 2 0
0 8
โคโฆ = ๐,
which gives the Smith normal form ๐ = ๐โ1๐ด๐ โ1 of ๐ด. We have now found a base๐โฒ1, ๐
โฒ2 of ๐ฟ in which each basis vector is a multiple of the corresponding basis vector
๐โฒ1, ๐โฒ2. Equivalently, the matrix ๐ is in diagonal form, which means that with respect to
the basis ๐โฒ1, ๐โฒ2, the fundamental parallelepiped ฮ (๐โฒ1, ๐
โฒ2) is a rectangle:
๐ฝ โฉฮ (๐โฒ1, ๐โฒ2) =
{๐1๐
โฒ1 + ๐2๐
โฒ2
๐1 โ [0, 2), ๐2 โ [0, 8), ๐1, ๐2 โ Z
}.
Figure 4.2: By changing bases on ๐ฝ and ๐ฟ we can transform the parallelepiped ฮ (๐ด) intoa parallelepiped ฮ (๐โฒ1, ๐
โฒ2) whose generators ๐
โฒ1, ๐
โฒ2 are multiples of the ๐โฒ1, ๐
โฒ2 respectively.
The figure on the left show the bases ๐1, ๐2 of ๐ฝ and ๐1, ๐2 of ๐ฟ that we start out with.In the center figure, we pass from basis ๐1, ๐2 to ๐โฒ1, ๐
โฒ2, thereby lining up ๐โฒ1 and ๐1. In
the right-hand figure, we pass from ๐1, ๐2 to ๐โฒ1, ๐โฒ2 thereby lining up ๐โฒ2 and ๐โฒ2. This
process corresponds to the computation of the Smith normal form of ๐ด.
We have now found a simple description of ๐ฝ โฉฮ (๐โฒ1, ๐โฒ2). How can we transform this
lattice-point set into the set ๐ฝ โฉ ฮ (๐1, ๐2) that we are interested in? Since both ๐1, ๐2and ๐โฒ1, ๐
โฒ2 are bases of ๐ฟ, the fundamental parallelepipeds contain the same number of
lattice points. However there is no linear transformation that gives a bijection betweenthese lattice-point sets.
For any basis ๐ด of ๐ฟ, the fundamental parallelepiped ฮ (๐ด) tiles the plane. Thismeans that for any basis ๐ด and any point ๐ง โ Z2 there is a unique ๐ฆ โ ๐ฟ such that๐งโ๐ฆ โ ฮ (๐ด). If we can find a formula for this map ๐ : ๐ง โฆโ ๐งโ๐ฆ for any given ๐ด, we canuse this map to transform ฮ (๐โฒ1, ๐
โฒ2) into ฮ (๐1, ๐2). Fortunately this is straightforward.
We simply find the corrdinates ๐ of ๐ง with respect to the basis ๐ด, i.e., we write ๐ง =๐1๐1 + ๐2๐2. If we now take fractional parts, we obtain
๐(๐ง) = {๐1}๐1 + {๐2}๐2 โ ฮ (๐1, ๐2).
104 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
Note that ๐ฆ = ๐ง โ ๐(๐ง) = โ๐1โ ๐1 + โ๐2โ ๐2 โ ๐ฟ. Using matrices and the fractional partfunction, ๐ can be described simply via ๐ = ๐ด โ frac โ๐ดโ1. This function ๐ is illustratedin Figure 4.3.
Figure 4.3: The lattice points in the fundamental parallelepiped ฮ (๐โฒ1, ๐โฒ2) can be
mapped into ฮ (๐ด) by means of the map ๐ = ๐ด โ frac โ๐ดโ1.
On the whole, this means that we can enumerate the lattice points in the fundamentalparallelepiped we are interested in via
๐ฝ โฉฮ (๐1, ๐2) = ๐ด โ frac โ๐ดโ1(๐ฝ โฉฮ (๐โฒ1, ๐โฒ2))
={(๐ด โ frac โ๐ดโ1 โ ๐)๐
๐ โ Z2 โฉ ([0, 2)ร [0, 8))
}where
๐ดโ1 โ ๐ =
โกโฃ 18 โ3
8
18
18
โคโฆ .
โกโฃ 1 0
โ1 1
โคโฆ =
โกโฃ 12 โ3
8
0 18
โคโฆ .
General Method
We now describe the method illustrated in the above example in full detail.Let ๐ฝ โ R๐ be a lattice and ๐ฟ a sublattice. Most of the time, we will have ๐ฝ = Z๐. Let
๐1, . . . , ๐๐ be a basis for ๐ฟ and let ๐ด denote the ๐ร๐-matrix with the ๐๐ as columns. Weare then interested in describing the set of ๐ฝ-lattice points contained in the fundamentalparallelepiped ฮ (๐1, . . . , ๐๐) of ๐ฟ with respect to the generators ๐1, . . . , ๐๐.
The starting point for this method is the observation that ฮ (๐ด) is easy to describeif the basis of ๐ฟ is a multiple of a basis of ๐ฝ .
Lemma 17. Let ๐ฝ be a lattice with basis ๐1, . . . , ๐๐ and let ๐ฟ be a sublattice with basis๐1, . . . , ๐๐ for ๐ โค ๐. Let ๐ผ โ [๐]. If ๐๐ = ๐ผ๐๐๐ for ๐ = 1, . . . , ๐ and ๐ผ๐ โ Zโฅ1, then the
4.3. IMPROVEMENTS BASED ON GEOMETRY 105
set of ๐ฝ-lattice points in the fundamental parallelepiped ฮ ๐ผ(๐ด) of ๐ฟ with respect to thebasis ๐ด is
๐ฝ โฉฮ ๐ผ(๐ด) =
{๐โ
๐=1
๐๐๐๐
๐๐ โ Z โ๐ โ [๐], 0 โค ๐๐ < ๐ผ๐ โ๐ โ ๐ผ, 0 < ๐๐ โค ๐ผ๐ โ๐ โ ๐ผ
}
and its generating function has the rational function representation
ฮฆฮ ๐ผ(๐ด)(๐ง) =โ๐โ๐ผ
๐ง๐ ยท๐โ
๐=1
1โ ๐ง๐ผ๐๐๐
1โ ๐ง๐๐.
๏ฟฝ
Proof. This follows immediately from the definition of ฮ ๐ผ(๐ด) and the fact that
๐โ๐=1
1โ ๐ง๐ผ๐๐๐
1โ ๐ง๐๐=โ๐ฅโ๐
๐ง๐ฅ for ๐ =
{๐โ
๐=1
๐๐๐๐
๐๐ โ {0, . . . , ๐ผ๐ โ 1} โ๐
}.
Changing the basis of ๐ฟ changes the fundamental parallelepiped. However, the num-ber of lattice points in the fundamental parallelepiped remains the same. Moreover thereis a natural bijection between the lattice points in the fundamental parallelepipeds. Thisbijection is essentially given by the โfractional partโ or the โdivision with remainderโ.
To make this precise in the next lemma, we need some additional notation.For any real number ๐ฅ โ R and ๐ โ {0, 1} we define integ๐(๐ฅ) and fract๐(๐ฅ) to be the
unique real numbers such that
๐ฅ = integ๐(๐ฅ) + fract๐(๐ฅ)
and
integ๐(๐ฅ) โ Z, 0 โค fract0(๐ฅ) < 1, 0 < fract1(๐ฅ) โค 1.
Note that integ0 ๐ฅ = โ๐ฅโ and fract0 ๐ฅ = {๐ฅ}. For any vector of real numbers ๐ฃ โ R๐ andany set ๐ผ โ [๐], we also write integ๐ผ ๐ฃ and fract๐ผ ๐ฃ to denote the vectors
integ๐ผ(๐ฃ) = (integ๐โ๐ผ(๐ฃ๐))๐โ[๐] and fract๐ผ(๐ฃ) = (fract๐โ๐ผ(๐ฃ๐))๐โ[๐]
where we interpret the exponent ๐ โ ๐ผ to denote 1 if ๐ โ ๐ผ and to denote 0 if ๐ โ ๐ผ.In analogy to the above notation we define an extension of the mod function. Let
๐, ๐ โ R. We define ๐ mod0 ๐ to be the unique real number 0 โค ๐ mod0 ๐ < ๐ suchthat ๐ = ๐๐+ ๐ mod0 ๐ for some ๐ โ Z. And we define ๐ mod1 ๐ to be the unique realnumber 0 < ๐ mod1 ๐ โค ๐ such that ๐ = ๐๐+ ๐ mod1 ๐ for some ๐ โ Z. Again, we willwrite mod๐โ๐ผ to denote mod0 if ๐ โ ๐ผ and mod1 if ๐ โ ๐ผ, etc. mod without exponentis understood to denote mod0 . Note that fract๐(๐๐ ) =
๐ mod๐ ๐๐ with this notation.
106 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
Lemma 18. Let ๐ฝ be a lattice in R๐ with basis ๐1, . . . , ๐๐ and let ๐ฟ be a sublattice. Let๐1, . . . , ๐๐ be a basis of ๐ฟ and let ๐ด denote the corresponding ๐ ร ๐-matrix. Let ๐ดโ1
denote any left inverse of ๐ด, i.e., let ๐ดโ1 be a matrix with the property ๐ดโ1๐ง = ๐ forany ๐ง โ span(๐ฟ) with ๐ง =
โ๐๐=1 ๐๐๐๐. Let ๐ผ โ [๐] and let ๐ โ Q๐.
1. For all ๐ง โ ๐ฝโฉ(๐+span(๐ฟ)) there exists a unique lattice point ๐ฆ โ ฮ ๐ผ(๐ด, ๐) such that
๐งโ๐ฆ โ ๐ฟ. Let ๐ง = ๐+โ๐
๐=1 ๐๐๐๐ and let ๐ = (๐๐)๐=1,...,๐. Then ๐ฆ = ๐+๐ด(fract๐ผ(๐)).
2. ๐ฝ โฉฮ ๐ผ(๐ด, ๐) is empty if and only if ๐ฝ โฉ (๐ + span(๐ฟ)) is empty.
3. Suppose ๐ โ ๐ฝ โฉ (๐ + span(๐ฟ)). Let ๐1, . . . , ๐๐ be a second basis of ๐ฟ and let ๐ตdenote the corresponding ๐ร ๐-matrix and let ๐ผ โฒ โ [๐]. Then
๐ฝ โฉฮ ๐ผ(๐ด, ๐) = ๐ด(fract๐ผ(๐ดโ1((๐ฝ โฉฮ ๐ผโฒ(๐ต))โ (๐ โ ๐)))) + ๐
and the above map induces a bijection between the sets ๐ฝโฉฮ ๐ผ(๐ด, ๐) and ๐ฝโฉฮ ๐ผโฒ(๐ต).
๏ฟฝ
Note that the map ๐ด โ fract๐ผ โ๐ดโ1 is not linear, as we have seen in the example.
Proof. 1. Let ๐ง โ ๐ฝ โฉ (๐+ span(๐ฟ)). Then there exist uniquely determined ๐๐ such that
๐ง = ๐ +๐โ
๐=1
๐๐๐๐ = ๐ +๐โ
๐=1
integ๐โ๐ผ(๐๐)๐๐ +๐โ
๐=1
fract๐โ๐ผ(๐๐)๐๐.
Notice that๐โ
๐=1
integ๐โ๐ผ(๐๐)๐๐ โ ๐ฟ โ ๐ฝ
and
๐ฆ = ๐ +๐โ
๐=1
fract๐ผ(๐๐)๐๐ = ๐ +๐ด(fract๐ผ(๐)) โ ฮ ๐ผ(๐ด, ๐).
Moreover, ๐ฆ โ ๐ฝ because ๐ง โ ๐ฝ . Notice also that ๐ฆ is uniquely determined as changingany coefficient by an integer amount takes us out of ฮ ๐ผ(๐ด, ๐).
2. By the first part of the theorem, we already know that if there exists a ๐ง โ๐ฝ โฉ (๐ + span(๐ฟ)), then there exists a ๐ฆ โ ๐ฝ โฉ ฮ ๐ผ(๐ด, ๐). Conversely, if ๐ฆ โ ๐ฝ โฉ ฮ ๐ผ(๐ด, ๐)then ๐ฆ โ ๐ฝ โฉ (๐ + span(๐ฟ)) because ฮ ๐ผ(๐ด, ๐) โ ๐ + span(๐ฟ).
3. Let ๐1 denote the translation ๐1(๐ฅ) = ๐ฅโ (๐โ๐) and let ๐2 denote the translation๐2(๐ฅ) = ๐ฅ+ ๐. Using this notation we have to show that the map
๐ = ๐2 โ๐ด โ fract๐ผ โ๐ดโ1 โ ๐1
gives a bijection from ๐ฝ โฉฮ ๐ผโฒ(๐ต) to ๐ฝ โฉฮ ๐ผ(๐ด, ๐). We will proceed in five steps.Step 1: The image of ๐ฝ โฉ span(๐ฟ) under ๐ is contained in ๐ฝ โฉ ฮ ๐ผ(๐ด, ๐). Let ๐ง โ
๐(๐ฝ โฉ (๐ + span(๐ฟ))). Then ๐ง has the form ๐ง = ๐ +โ๐
๐=1 fract๐ผ(๐๐)๐๐ where the ๐๐ are
4.3. IMPROVEMENTS BASED ON GEOMETRY 107
such thatโ๐
๐=1 ๐๐๐๐ โ (๐ฝ โฉ span๐ฟ) โ (๐ โ ๐). Therefore ๐ง is contained in ฮ ๐ผ(๐ด, ๐) andit remains to show ๐ง โ ๐ฝ . By construction, the coefficients ๐๐ have the property
๐โ๐=1
๐๐๐๐ = โ๐โ
๐=1
๐๐๐๐ +๐โ
๐=1
๐๐๐๐
where ๐ โ ๐ =โ๐
๐=1 ๐๐๐๐ andโ๐
๐=1 ๐๐๐๐ โ ๐ฝ . Therefore
๐ง = ๐ +๐โ
๐=1
fract๐ผ(๐๐)๐๐
= ๐+ (๐ โ ๐) +๐โ
๐=1
fract๐ผ(๐๐)๐๐
= ๐+
๐โ๐=1
๐๐๐๐ +
๐โ๐=1
fract๐ผ(โ๐๐ + ๐๐)๐๐
= ๐+๐โ
๐=1
๐๐๐๐ +๐โ
๐=1
(โ๐๐ + ๐๐)๐๐ โ๐โ
๐=1
integ๐ผ(โ๐๐ + ๐๐)๐๐
= ๐+๐โ
๐=1
๐๐๐๐ โ๐โ
๐=1
integ๐ผ(โ๐๐ + ๐๐)๐๐.
Now ๐ andโ๐
๐=1 ๐๐๐๐ are elements of ๐ฝ by assumption andโ๐
๐=1 integ๐ผ(โ๐๐+๐๐)๐๐ is an
element of ๐ฝ because the coefficients are integer and ๐ฟ is a sublattice of ๐ฝ . Thus ๐ง โ ๐ฝ .Step 2: If ๐ข, ๐ฃ โ ๐ฝ โฉ span(๐ฟ) and ๐(๐ข) = ๐(๐ฃ), then ๐ข โ ๐ฃ โ ๐ฟ. Let ๐ข =
โ๐๐=1 ๐๐๐๐
and ๐ฃ =โ๐
๐=1 ๐๐๐๐. Let ๐ โ ๐ =โ๐
๐=1 ๐ ๐๐๐. Then ๐(๐ข) = ๐(๐ฃ) is equivalent to
๐โ๐=1
fract๐ผ(๐๐ โ ๐ ๐)๐๐ =๐โ
๐=1
fract๐ผ(๐๐ โ ๐ ๐)๐๐.
Then
๐ขโ ๐ฃ =
๐โ๐=1
(๐๐ โ ๐ ๐)๐๐ โ๐โ
๐=1
(๐๐ โ ๐ ๐)๐๐
=๐โ
๐=1
fract๐ผ(๐๐ โ ๐ ๐)๐๐ โ๐โ
๐=1
fract๐ผ(๐๐ โ ๐ ๐)๐๐ +๐โ
๐=1
integ๐ผ(๐๐ โ ๐ ๐)๐๐ โ๐โ
๐=1
integ๐ผ(๐๐ โ ๐ ๐)๐๐
=๐โ
๐=1
integ๐ผ(๐๐ โ ๐ ๐)๐๐ โ๐โ
๐=1
integ๐ผ(๐๐ โ ๐ ๐)๐๐
which is an integral combination of basis vectors of ๐ฟ and therefore contained in ๐ฟ asclaimed.
108 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
Step 3: ๐ |๐ฝโฉฮ ๐ผโฒ (๐ต) is injective. As ๐ต is a basis of ๐ฟ, no two distinct elements of
๐ฝโฉฮ ๐ผโฒ(๐ต) differ by an element of ๐ฟ. Therefore Step 2 implies that ๐ |๐ฝโฉฮ ๐ผโฒ (๐ต) is injective.
Step 4: Both ๐ฝ โฉ ฮ ๐ผโฒ(๐ต) and ๐ฝ โฉ ฮ ๐ผ(๐ด, ๐) contain the same finite number of latticepoints. This follows directly from the fact that ๐ต and ๐ด are bases of the same lattice.
Step 5: ๐ |๐ฝโฉฮ ๐ผโฒ (๐ต) is bijective. Combining Steps 3 and 4 yields bijectivity.
The previous lemma gives us a way to keep track of ฮ ๐ผ(๐ด, ๐) when we change basesof ๐ฟ. Handling changes of bases of ๐ฝ is far easier, as the next lemma tells us.
Lemma 19. If ๐ is a lattice transformation of the lattice ๐ฝ , then
๐ฝ โฉฮ (๐ด; ๐) = ๐ (๐ฝ โฉฮ (๐โ1๐ด;๐โ1๐)).
๏ฟฝ
Proof. This follows from ๐ฝ = ๐๐ฝ = ๐โ1๐ฝ and ๐ด๐ + ๐ = ๐ (๐โ1๐ด๐ + ๐โ1๐) for all๐ โ [0, 1)๐.
So far, Lemma 17 tells us that lattice-point sets in โrectangularโ parallelepipeds areeasy to describe and Lemmas 18 and 19 allow us to control ๐ฝ โฉฮ ๐ผ(๐ด, ๐) when we changebases on ๐ฟ or ๐ฝ , respectively. So what is left to do is to find a way to transform anarbitrary fundamental parallelepiped ฮ ๐ผ(๐ด, ๐) into a โrectangularโ parallelepiped usingthe transformations given in Lemmas 18 and 19. To this end we are going to use theSmith normal form as illustrated with the above example.
Lemma 20. Let ๐ฝ = Z๐ be a lattice and ๐ฟ a sublattice of ๐ฝ with basis ๐1, . . . , ๐๐ andcorresponding ๐ร ๐-matrix ๐ด. Let ๐ โ R๐. If ๐ + span(๐ฟ) does not contain any integerpoint, then Z๐ โฉฮ ๐ด,๐ is empty. Otherwise, let ๐ โ Z๐ โฉ span(๐ฟ). Then
Z๐ โฉฮ ๐ผ(๐ด, ๐) = ๐ด โ fract๐ผ โ๐ โ1 โ ๐โ1(Z๐ โฉฮ (๐)โ ๐โ1(๐ โ ๐)) + ๐
where ๐ด = ๐๐๐ is the Smith normal form of ๐ด, the numbers ๐ 1, . . . , ๐ ๐ are the diagonalentries of ๐ ร ๐-matrix ๐ and ๐โ1 is the diagonal ๐ ร ๐-matrix with diagonal entries1๐ 1, . . . , 1
๐ 1. ๏ฟฝ
Note that in the above lemma
Z๐ โฉฮ (๐) = Z๐ โฉ [0, ๐ 1)ร . . .ร [0, ๐ ๐)ร {0} ร . . .ร {0},
is a rectangle of lattice points.
Proof. In order to be able to apply Lemma 18 we need to pick a left inverse ๐ดโ1 of ๐ด.To this end we define ๐ดโ1 := ๐ โ1๐โ1๐โ1. As ๐โ1๐ = ๐ผ is the identity, ๐ดโ1 is indeeda left inverse of ๐ด since ๐ดโ1๐ด = ๐ โ1๐โ1๐โ1๐๐๐ = ๐ผ. We then have
Z๐ โฉฮ ๐ผ(๐ด, ๐) = ๐ด โ fract๐ผ โ๐ดโ1((Z๐ โฉฮ (๐ด๐ โ1))โ (๐ โ ๐)) + ๐
= ๐ด โ fract๐ผ โ๐ดโ1 โ ๐((Z๐ โฉฮ (๐โ1๐ด๐ โ1))โ ๐โ1(๐ โ ๐)) + ๐
= ๐ด โ fract๐ผ โ๐ดโ1 โ ๐((Z๐ โฉฮ (๐))โ ๐โ1(๐ โ ๐)) + ๐
= ๐ด โ fract๐ผ โ๐ โ1 โ ๐โ1((Z๐ โฉฮ (๐))โ ๐โ1(๐ โ ๐)) + ๐
4.3. IMPROVEMENTS BASED ON GEOMETRY 109
where the first identity follows from Lemma 18 using the fact that ๐ด๐ โ1 is a latticebasis for ๐ฟ, the second identity follows from Lemma 19 using the fact that ๐ is a latticetransformation, the third identitiy follows from ๐ด = ๐๐๐ and the last identity followsfrom our choice of ๐ดโ1 and ๐โ1.
Theorem 4.2Let ๐1, . . . , ๐๐ โ Z๐ be linearly independent and let ๐ด = (๐๐๐)๐โ[๐],๐โ[๐] be the corre-sponding matrix with Smith normal form ๐ด = ๐๐๐ . Let ๐ โ R๐ and let ๐ผ โ [๐]. Let๐ 1, . . . , ๐ ๐ denote the diagonal entries of ๐ and define ๐ก๐ =
๐ ๐๐ ๐. Note that ๐ก๐ โ Z as ๐ ๐|๐ ๐+1.
Let ๐ โ1 = (๐ฃ๐๐)๐๐ where ๐ is the row-index and ๐ the column index. If ๐ โ Z๐ โฉ span(๐ด),then the generating function of the set of lattice points in the fundamental parallelepipedฮ ๐ผ(๐ด, ๐) is given by
ฮฆ๐Z(๐ด;๐)(๐ฅ) =
๐ 1โ1โ๐1=0
ยท ยท ยท๐ ๐โ1โ๐๐=0
๐ฅ
(1๐ ๐
โ๐๐=1 ๐๐๐(
โ๐๐=1(๐๐โ๐ค๐)๐ฃ๐๐๐ก๐ mod๐โ๐ผ ๐ ๐)+๐๐
)๐=1,...,๐ ,
where ๐ค = ๐โ1(๐ โ ๐). ๏ฟฝ
Proof. From Lemmas 17 and 20 we immediately get
๐ 1โ1โ๐1=0
ยท ยท ยท๐ ๐โ1โ๐๐=0
๐ฅ๐ด fract๐ผ ๐ โ1๐โ1((๐1,...,๐๐,0,...,0)๐กโ๐ค)+๐.
We then calculate
๐ โ1๐โ1((๐1, . . . , ๐๐, 0, . . . , 0)๐ก โ ๐ค) =
(๐โ
๐=1
๐๐ โ ๐ค๐
๐ ๐๐ฃ๐๐
)๐=1,...,๐
=1
๐ ๐
(๐โ
๐=1
(๐๐ โ ๐ค๐)๐ก๐๐ฃ๐๐
)๐=1,...,๐
where we denote the entries of ๐ โ1 by ๐ฃ๐๐. Recall that for ๐ โ {0, 1} we havefract๐(๐๐ ) =
๐ mod๐ ๐๐ . With this notation we get
fract๐ผ
(1
๐ ๐
๐โ๐=1
(๐๐ โ ๐ค๐)๐ก๐๐ฃ๐๐
)๐=1,...,๐
=
(1
๐ ๐
๐โ๐=1
(๐๐ โ ๐ค๐)๐ก๐๐ฃ๐๐ mod๐โ๐ผ ๐ ๐
)๐=1,...,๐
.
Finally, we calculate
๐ด
(1
๐ ๐
โ๐
(๐๐ โ ๐ค๐)๐ฃ๐๐๐ก๐ mod๐โ๐ผ ๐ ๐
)๐=1,...,๐
+ ๐ =
โโ 1
๐ ๐
๐โ๐=1
๐๐๐
(๐โ
๐=1
(๐๐ โ ๐ค๐)๐ฃ๐๐๐ก๐ mod๐โ๐ผ ๐ ๐
)+ ๐๐
โโ ๐=1,...,๐
.
110 CHAPTER 4. PARTITION ANALYSIS VIA POLYHEDRAL GEOMETRY
4.4 Conclusions
One should note that the iterative algorithm is in the intersection of MacMahonโs ap-proach and polyhedral geometry. During the execution of the algorithm, geometricobjects such as the vertices and the vertex cones of the ฮฉ-polyhedron, as well as a tri-angulation of the duals of the vertex cones, are computed, without use of any tool frompolyhedral geometry. Moreover, by using the intermediate step of computing cones sym-bolically, some of the cones are dropped without computing their generating functions,i.e., if all generators of a cone are pointing down and the apex lies below the intersectinghalfspace, the cone can be safely ignored.
On the other hand, the algorithm for simultaneous multiple ๐ elimination leansclearly towards polyhedral geometry, using exclusively tools from that area and ignoringthe recursive ๐ elimination, prominent in the traditional partition analysis implemen-tations. In the polyhedral geometry world, already for a decade, there exist tools, likethe algorithm of Barvinok-Woods [13] for computing Hadamard products, that can beapplied to perform simultaneous ๐ elimination. Nevertheless, in the partition analysisworld these tools are not yet widespread and they are not tested concerning possibleadvantages, either practical or theoretical.
A practical implementation of the algorithms and further theoretical developmentsare under preparation and will be presented in [19].
Appendix A
Proof of โGeometry of Omega2โ
In what follows we assume gcd(๐ผ, ๐ฝ) = 1 as indicated in Theorem 3.1.
Fundamental Parallelepipeds
Let ฮ ๐ = ฮ ๐ด โฉ (Z2 ร {๐}) and ฮ๐ = ฮ ๐ต โฉ (Z2 ร {๐}) for ๐ โ Z. ฮ ๐ (resp. ฮ๐) is theintersection of ฮ ๐ด (resp. ฮ ๐ต ) with the affine subspace {(๐ฅ, ๐ฆ, ๐) โ R3}.
Lemma 21. There is at most one lattice point in each fundamental parallelepiped ofthe cones ๐ด and ๐ต at any given height (๐ฅ3-value) and if there is one then its form isdetermined as follows:
โ ฮ ๐ = {(๐ฅ1, ๐ฅ2, ๐) โ Z3|๐ฅ1 = ๐โ๐ฅ2๐ฝ๐ผ , ๐ฅ2 โ {0, 1, . . . , ๐ผโ 1}} .
โ ฮ๐ = {(๐ฅ1, ๐ฅ2, ๐) โ Z3|๐ฅ2 = ๐โ๐ฅ1๐ผ๐ฝ , ๐ฅ1 โ {โ๐ฝ + 1,โ๐ฝ + 2, . . . , 0}}.
Moreover |ฮ ๐ | โค 1 and |ฮ๐ | โค 1. ๏ฟฝ
Proof. Let (๐ฅ1, ๐ฅ2, ๐ฅ3) โ ฮ ๐ด โฉ Z3. Then there exist ๐, ๐ โ [0, 1) such that
๐(โ๐ฝ, ๐ผ, 0) + ๐(1, 0, ๐ผ) = (๐ฅ1, ๐ฅ2, ๐ฅ3) โ Z3.
This translates to the systemโงโชโจโชโฉ๐ฅ1 = ๐ โ ๐๐ฝ
๐ฅ2 = ๐๐ผ
๐ฅ3 = ๐๐ผ
โ
โงโชโจโชโฉ๐ฅ1 =
๐๐ผ โ ๐๐ฝ
๐ฅ2 โ [0, ๐ผ) โฉ Z๐ = ๐ฅ3
๐ผ
โ
โงโชโจโชโฉ๐ฅ1 =
๐ฅ3โ๐ฅ2๐ฝ๐ผ
๐ฅ2 โ {0, 1, . . . , ๐ผโ 1}๐ฅ3 โ {0, 1, . . . , ๐ผโ 1}
Fix ๐ฅ3 = ๐ โ {0, 1, . . . , ๐ผ โ 1}. Assume |ฮ ๐ | > 1 and let (๐ฅโฒ1, ๐ฅโฒ2, ๐), (๐ฅ
โฒโฒ1, ๐ฅ
โฒโฒ2, ๐) โ ฮ ๐ .
Then {๐ผ|๐ โ ๐ฅโฒ2๐ฝ
๐ผ|๐ โ ๐ฅโฒโฒ2๐ฝโ ๐ผ|๐ฅโฒ2๐ฝ โ ๐ฅโฒโฒ2๐ฝ โ
{๐ผ|๐ฝ(๐ฅโฒ2 โ ๐ฅโฒโฒ2)
gcd(๐ผ, ๐ฝ) = 1โ ๐ผ|๐ฅโฒ2 โ ๐ฅโฒโฒ2
Since |๐ฅโฒ2 โ ๐ฅโฒโฒ2| < ๐ผ we have a contradiction. Thus |ฮ ๐ | โค 1.The proof for ฮ๐ is completely analogous.
111
112 APPENDIX A. PROOF OF โGEOMETRY OF OMEGA2โ
The first summand
Our goal is to show that ๐๐ด =๐๐ผ,๐ฝ
(1โ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ๐ง1๐ง๐ผ3 )
.
Proposition 4.
๐ฮ (๐ด) = ๐๐ผ,๐ฝ =
๐ผโ1โ๐=0
๐๐๐ง๐3
for
๐๐ =
โงโจโฉ๐ง๐๐ฝ
2 if ๐ฝ|๐ or ๐ = 0,
๐งrmd((๐ผโ1 mod ๐ฝ)๐,๐ฝ)โ๐ฝ1 ๐ง
rmd((๐ฝโ1 mod ๐ผ)๐,๐ผ)2 otherwise,
๏ฟฝ
Proof. Since ฮ ๐ = โ for ๐ โฅ ๐ผ, we need to prove the following three statements
1. ๐ฮ ๐= 1 = ๐0.
2. Let ๐ โ {1, 2, . . . , ๐ผโ 1} such that ๐ฝ|๐. Then ๐ฮ ๐= ๐ง
๐๐ฝ
2 ๐ง๐3 = ๐๐๐ง๐3 .
3. Let ๐ โ {1, 2, . . . , ๐ผโ 1} such that ๐ฝ - ๐. Then ๐ฮ ๐= ๐๐๐ง
๐3 .
Using the equations from the proof of Lemma 21 we haveโงโชโจโชโฉ๐ฅ1 = ๐ โ ๐๐ฝ
๐ฅ2 = ๐๐ผ
0 = ๐๐ผ
๐=0โโโ
{๐๐ฝ โ Z๐๐ผ โ Z
โ๐โNโโโโ
{๐๐ฝ โ Z๐ = ๐
๐ผ
โโ
{๐๐ฝ๐ผ โ Z๐ = ๐
๐ผ
โโ
{๐ผ|๐ or ๐ผ|๐ฝ๐ = ๐
๐ผ
gcd(๐ผ,๐ฝ)=1โโโโโโโโ
{๐ผ|๐๐๐ผ = ๐ โ [0, 1)
Thus ๐ = 0, which means ๐ฅ1 = ๐ฅ2 = ๐ฅ3 = 0 and ๐ฮ 0= 1.
By the definition of ๐0 we have that ๐0 = 1.๏ฟฝ of statement 1.
From Lemma 21 we know that |ฮ ๐ | โค 1 and if equality holds the lattice point is
of the form ( ๐โ๐ฅ2๐ฝ๐ผ , ๐ฅ2, ๐) for some ๐ฅ2 โ {0, 1, . . . , ๐ผ โ 1}. We will check if the desired
exponent is actually a lattice point in ฮ ๐ . Let ๐ฅ2 = ๐๐ฝ . Since ๐ < ๐ผ we have that
๐ฅ2 โ {0, 1, . . . , ๐ผโ 1}. Moreover
๐ฅ1 =๐ โ ๐ฅ2๐ฝ
๐ผ=
๐ โ ๐๐ฝ๐ฝ
๐ผ= 0 โ Z
which means that (0, ๐๐ฝ , ๐) โ ฮ ๐ . Thus ๐ฮ ๐
= ๐ง๐๐ฝ
2 ๐ง๐3 , which by definition is ๐๐๐ง๐3 .
๏ฟฝ of statement 2.
113
We will proceed in two steps. First show that rmd((๐ฝโ1 mod ๐ผ)๐, ๐ผ) is the ๐ฅ2-coordinate of a lattice point in ฮ ๐ and then that rmd((๐ผโ1 mod ๐ฝ)๐, ๐ฝ) โ ๐ฝ is the ๐ฅ1-
coordinate of a lattice point in ฮ ๐ . Since |ฮ ๐ | โค 1, we have that ๐ฮ ๐= ๐๐๐ง
๐3 .
โ In order to prove that rmd((๐ฝโ1 mod ๐ผ)๐, ๐ผ) is the ๐ฅ2-coordinate of a lattice pointin ฮ ๐ we have to show that
1. rmd((๐ฝโ1 mod ๐ผ)๐, ๐ผ) โ {0, 1, . . . , ๐ผโ 1};
2.๐โ(rmd((๐ฝโ1 mod ๐ผ)๐,๐ผ))๐ฝ
๐ผ โ Z.
Let ๐ฅ2 = rmd((๐ฝโ1 mod ๐ผ)๐, ๐ผ). By definition the remainder of division by ๐ผ is in{0, 1, . . . , ๐ผโ 1}.By the definitions of remainder and modular inverse we have
๐ฅ2 = ๐๐ โ ๐ผ๐
for some ๐ โ Z and ๐ such that ๐๐ฝ = ๐ผ๐+ 1 for some ๐ โ Z.Then
๐ฅ2 = ๐๐ โ ๐ผ๐โ ๐ฝ๐ฅ2 = ๐ฝ๐๐ โ ๐ผ๐ฝ๐
โ ๐ฝ๐ฅ2 = (๐ผ๐+ 1)๐ โ ๐ผ๐ฝ๐
โ ๐ฝ๐ฅ2 = ๐ผ๐๐ + ๐ โ ๐ผ๐ฝ๐
โ ๐ โ ๐ฝ๐ฅ2 = ๐ผ(๐ฝ๐โ ๐๐)
โ ๐ โ ๐ฝ๐ฅ2๐ผ
= ๐ฝ๐โ ๐๐ โ Z.
โ In order to prove that rmd((๐ผโ1 mod ๐ฝ)๐, ๐ฝ) โ ๐ฝ is the ๐ฅ1-coordinate of a latticepoint in ฮ ๐ we have to show that there exist ๐ฅ2 โ {0, 1, . . . , ๐ผ โ 1} such that
rmd((๐ผโ1 mod ๐ฝ)๐, ๐ฝ)โ ๐ฝ = ๐โ๐ฅ2๐ฝ๐ผ .
Let ๐ = rmd((๐ผโ1 mod ๐ฝ)๐, ๐ฝ). By definition
๐ = ๐๐ โ ๐ฝ๐
for some ๐ โ Z and ๐ such that ๐ผ๐ = ๐ฝ๐ + 1 for some ๐ โ Z. Then we want toprove that
๐โ ๐ฝ =๐ โ ๐ฅ2๐ฝ
๐ผ.
We have
๐โ ๐ฝ =๐ โ ๐ฅ2๐ฝ
๐ผโ ๐๐ โ ๐ฝ๐โ ๐ฝ =
๐ โ ๐ฅ2๐ฝ
๐ผโ ๐ผ๐๐ โ ๐ผ๐ฝ๐โ ๐ผ๐ฝ = ๐ โ ๐ฅ2๐ฝ
โ ๐ฝ๐๐ + ๐ โ ๐ผ๐ฝ๐โ ๐ผ๐ฝ = ๐ โ ๐ฅ2๐ฝ
โ ๐ฅ2 = ๐ผ(๐+ 1)โ ๐๐.
114 APPENDIX A. PROOF OF โGEOMETRY OF OMEGA2โ
Thus we need to show that ๐ผ(๐ + 1) โ ๐๐ โ {0, 1, . . . , ๐ผ โ 1}. First observe that๐ผ(๐+ 1)โ ๐๐ โ Z.If ๐ is the quotient in the division of ๐๐ by ๐ฝ we have ๐ + 1 > ๐๐
๐ฝ . In order toshow that ๐ผ(๐+ 1)โ ๐๐ > 0 we have
๐+ 1 >๐๐
๐ฝโ ๐ >
๐๐
๐ฝโ 1
โ ๐ >
(๐ฝ๐+1๐ผ
)๐
๐ฝโ 1
โ ๐ >๐ฝ๐๐ + ๐
๐ผ๐ฝโ 1
โ ๐ >๐ฝ๐๐
๐ผ๐ฝโ 1
โ ๐ >๐๐
๐ผโ 1
โ ๐ผ(๐+ 1) > ๐๐
โ ๐ผ(๐+ 1)โ ๐๐ > 0.
For the other direction, i.e. ๐ผ(๐+1)โ๐๐ < ๐ผ we observe that since ๐ฝ - ๐ we have0 < ๐ = ๐๐ โ๐๐ฝ. We know that ๐ < ๐ผ and
๐ < ๐ผ๐โ ๐
๐ผ< ๐
โ 0 < ๐โ ๐
๐ผ
โ ๐๐ฝ < ๐๐ฝ + ๐โ ๐
๐ผ
โ ๐๐ฝ < ๐๐ โ ๐
๐ผ
โ ๐๐ฝ <(๐๐ฝ + 1)๐
๐ผโ ๐
๐ผ
โ ๐๐ฝ <๐๐ฝ๐
๐ผ
โ ๐ <๐๐
๐ผโ ๐ผ๐ < ๐๐
โ ๐ผ๐+ ๐ผ < ๐๐ + ๐ผ
โ ๐ผ(๐+ 1)โ ๐๐ < ๐ผ.
๏ฟฝ of statement 3.
Corollary 1. From Proposition 4 and Lemma 4 we have that ๐๐ด =๐๐ผ,๐ฝ
(1โ๐งโ๐ฝ1 ๐ง๐ผ2 )(1โ๐ง1๐ง๐ผ3 )
.
๏ฟฝ
115
The second summand
We want to show that ๐๐ต =๐๐ผ,๐ฝ+1โ๐งโ๐ฝ
1 ๐ง๐ผ2(1โ๐งโ๐ฝ
1 ๐ง๐ผ2 )(1โ๐ง1๐ง๐ผ3 ).
Proposition 5.
๐ฮ (๐ต) = ๐๐ผ,๐ฝ + 1โ ๐งโ๐ฝ1 ๐ง๐ผ2 =
๐ฝโ1โ๐=0
๐๐๐ง๐3 + 1โ ๐งโ๐ฝ
1 ๐ง๐ผ2
for
๐๐ =
{๐งโ๐ฝ1 ๐ง๐ผ2 if ๐ = 0,
๐งrmd((๐ผโ1 mod ๐ฝ)๐,๐ฝ)โ๐ฝ1 ๐ง
rmd((๐ฝโ1 mod ๐ผ)๐,๐ผ)2 otherwise.
๏ฟฝ
Proof. Since ฮ๐ = โ for ๐ โฅ ๐ฝ, we need to prove the following three statements
1. ๐ฮ0(z) = 1 and ๐0 = ๐งโ๐ฝ
1 ๐ง๐ผ2 .
2. If ๐ โ {1, 2, . . . , ๐ฝ โ 1}, then ๐ฮ๐(z) = ๐๐๐ง
๐3 .
The proof is analogous to that of Proposition 4โงโชโจโชโฉ๐ฅ1 = โ๐๐ฝ๐ฅ2 = ๐ + ๐๐ผ
0 = ๐๐ฝ
๐=0โโโ
{๐๐ฝ โ Z๐๐ผ โ Z
โ๐โNโโโโ
{๐๐ฝ โ Z๐ = ๐
๐ผ
โโ
{๐๐ฝ๐ผ โ Z๐ = ๐
๐ผ
โโ
{๐ผ|๐ or ๐ผ|๐ฝ๐ = ๐
๐ผ
gcd(๐ผ,๐ฝ)=1โโโโโโโโ
{๐ผ|๐๐๐ผ = ๐ โ [0, 1)
Thus ๐ = 0, ๐ฅ1 = ๐ฅ2 = ๐ฅ3 = 0 and ๐ฮ0= 1.
By the definition of ๐0 we have that ๐0 = ๐งโ๐ฝ1 ๐ง๐ผ2 .
๏ฟฝ of statement 1.We need to show that ฮ ๐ = ฮ๐ for ๐ โ {1, 2, . . . , ๐ฝ โ 1}.From the analysis in the proof of Proposition 4, we know that
(rmd((๐ผโ1 mod ๐ฝ)๐, ๐ฝ)โ ๐ฝ, rmd((๐ฝโ1 mod ๐ผ)๐, ๐ผ), ๐) โ ฮ ๐
for ๐ โ {0, 1, . . . , ๐ฝ โ 1} โ {0, 1, . . . , ๐ผโ 1} .The proof follows from the fact that ๐ฅ1 =
๐โ๐ฅ2๐ฝ๐ผ โ ๐ฅ2 =
๐โ๐ฅ1๐ผ๐ฝ .
๏ฟฝ of statement 2.
Now the proof of Theorem 3.2 follows from Proposition 4, Proposition 5 and consid-ering the signs in the two summands of (3.10).
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Index
๐ generating function, 70ฮฉโฅ, algebraic84ฮฉโฅ operator, 69Hilbert-Poincare series, 83linear Diophantine system, 52truncated multivariate Hilbert-Poincare se-
ries, 86
Complete Homogeneous Symmetric Poly-nomials, 77
conefeasible, 29half-open, 28open, 28pointed, 24polyhedral, 22rational, 22simplicial, 23tangent, 29triangulation, 29unimodular, 24vertex, 29
conic hull, 22crude generating function, 70
dimensionpolyhedron, 18
Elliottโs Algorithm, 65
face, 20facet, 20forward direction, 46fundamental parallelepiped, 23Fundamental Recurrence, 77
generating function, 35, 36
rational, 36
halfspace, 16
hull
affine, 18
conic, 22
hyperplane, 16
affine, 16
supporting, 20
indicator function, 35
integral
closure, 26
element, 26
integral domain, 30
lattice, 25
Laurent polynomials, 30
Laurent series
multivariate, 31
univariate, 31
linear Diophantine problem
counting, 54
listing, 54
linear Diophantine systems
hierarchy, 56
linear functional, 16
MacMahon , rules72
monoid, 25
polyhedral cone, 22
polyhedron, 17
dimension, 18
rational, 17
polytope, 19
120
INDEX 121
rational function, 30Ray, 20recession cone, 46
semigroup, 25affine, 25generating set, 25saturated, 26
Symbolic Computation, 76
translation of cone, 27triangulation, 29
vector partition function, 52vector space
graded, 83vertex cone, 29
Curriculum Vitae
Personal details
Last Name Zafeirakopoulos
First Name Zafeirakis
Father Name Neoklis
Nationality Greek
Date of birth June 11, 1982
Email [email protected], [email protected]
Webpage www.zafeirakopoulos.info
Education
Title PhD in Technical Sciences (2009-present)
Johannes Kepler University
Department Research Institute for Symbolic Computation (RISC)
Advisor Prof. Peter Paule
Thesis title Linear Diophantine Systems:
Partition Analysis and Polyhedral Geometry
Title Masterโs in Computational Science (2006-2008)
University of Athens, Department of Informatics and Telecom
Advisor: Prof. Ioannis Emiris
Thesis title: Study and Benchmarks for Real Root Isolation methods
Grade: 8.72/10
123
124 INDEX
Title ฮ ฯฯ ฯฮฏฮฟฮฝ (4-years degree) in Mathematics (2000-2005)
University of Athens, Department of Mathematics
Grade: 7.2/10
Major: Applied Mathematics
Minor: Computational Mathematics
Languages
Greek (native), English (very good), German (basic), French (basic)
Awards
Marshall Plan Scholarship Jan-Jun 2012Research Visit to Prof. Matthias Beck (San Francisco State University),โAlgorithmic Solution of Linear Diophantine Systemsโ
Service and research activities
โ Reviewer for the conferences International Symposium on Experimental Algo-rithms 2011, Symbolic Numeric Computation 2011, SIAM Conference on Geo-metric and Physical Modeling 2011, International Symposium on Symbolic andAlgebraic Computation 2012 and the International Journal of Combinatorics.
โ Organizer for the seminars โIntroduction to Java - Scientific Programmingโ (2004,2005) and โOpen Source: Object Oriented Programming sessionโ (2004).
โ Lecturer in the summer school of the Nesin Mathematics Village (2010,2011,2012)
โ AIM SQuaRE (Nov 2010, 2011, 2012, Research Visit - American Institute of Math-ematics, Palo Alto, โPartition Theory and Polyhedral Geometryโ
โ SageDays (July 2010), RISC, โBenchmarksโ,โSpecial Functionsโ
โ SageDays 31 (June 2011), Washington, โPolyhedral Geometryโ, โSymbolicsโ
Publications
1. M. Hemmer, E.P. Tsigaridas, Z. Zafeirakopoulos, I.Z. Emiris, M.I. Karavelas, B.Mourrain, โExperimental evaluation and cross-benchmarking of univariate realsolversโ, Proceedings of the 2009 conference on Symbolic numeric computation,45โ54, 2009, ACM
2. H. Rahkooy, Z. Zafeirakopoulos, โUsing resultants for inductive Grobner basescomputationโ, ACM Communications in Computer Algebra, volume 45, number1/2, 135โ136, 2011, ACM
INDEX 125
3. C. Koukouvinos, D.E. Simos, Z. Zafeirakopoulos, โAn Algebraic Framework forExtending Orthogonal Designsโ, ACM Communications in Computer Algebra, vol-ume 45, number 1/2, 123โ124, 2011, ACM
4. C. Koukouvinos, V. Pillwein, D.E. Simos, and Z. Zafeirakopoulos, โOn the averagecomplexity for the verification of compatible sequencesโ, Information ProcessingLetters, 111, 17, 825โ830, 2011, Elsevier
5. M. Beck, B. Braun, M. Koeppe, C. Savage, Z. Zafeirakopoulos, โs-Lecture HallPartitions, Self-Reciprocal Polynomials, and Gorenstein Conesโ, arXiv preprintarXiv:1211.0258, 2012