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Combinatorial Aspects of Tropical Geometry

by

Josephine Thi Mar Lwin Yu

B.S. (University of California, Davis) 2003

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge:Professor Bernd Sturmfels, Chair

Professor Lior PachterProfessor Alper Atamturk

Spring 2007

The dissertation of Josephine Thi Mar Lwin Yu is approved:

Chair Date

Date

Date

University of California, Berkeley

Spring 2007


Copyright 2007

by


1

Abstract


by


Doctor of Philosophy in Mathematics

University of California, Berkeley

Professor Bernd Sturmfels, Chair

Tropical convex geometry and tropical algebraic geometry arise from linear and

polynomial algebra over the tropical semiring (R,min,+). They also appear as images of

convex and algebraic sets over fields with valuations into the real numbers. In this thesis,

the combinatorial aspects of tropical geometry are studied through three topics: tropical

polytopes, tropical linear spaces, and tropical elimination and implicitization.

Tropical polytopes are related to cellular resolutions of monomial ideals in two dif-

ferent ways. First, their natural polyhedral complex structure supports minimal linear free

resolutions of monomial initial ideals of the ideals of 2× 2-minors of matrices of unknowns.

Secondly, for an arbitrary monomial ideal, the tropical convex hull of the exponents of its

minimal generators is the common shadow of a natural family of cellular free resolutions,

generalizing the hull resolution.

Tropical linear spaces are some special types of intersections of tropical hyper-

2

planes. We show that they are also images of certain tropical linear maps. We characterize

the sets of tropical hyperplanes and the parametrizations that define a tropical linear space.

This generalizes the problem of finding minimal tropical bases of matroids. We show that

graphic and cographic matroids have unique minimal tropical bases.

Elimination and implicitization are computational problems of finding the defining

ideals of a projection of an algebraic variety and a variety given by a parametrization,

respectively. We give a combinatorial construction of the tropicalization of these ideals when

the given input polynomials have generic coefficients. When the solution is a principal ideal,

we can recover the Newton polytope of the generating polynomial from the tropical variety,

and in general we can compute its Chow polytope. We also describe an implementation of

this method.

Professor Bernd SturmfelsDissertation Committee Chair

i

To my parents.

ii

Contents

List of Figures iv

List of Tables vi

1 Introduction 1

1.1 Polyhedral geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Commutative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Tropical mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Tropical convex geometry . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Tropical linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.3 Tropical algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . 11

2 Tropical Polytopes and Cellular Resolutions 13

2.1 Tropical polytopes via cellular resolutions . . . . . . . . . . . . . . . . . . . 142.1.1 An algorithm for computing tropical convex hulls . . . . . . . . . . . 232.1.2 Tropical cyclic polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Cellular resolutions via tropical polytopes . . . . . . . . . . . . . . . . . . . 332.2.1 Faces of tropical polytopes . . . . . . . . . . . . . . . . . . . . . . . 40

3 Tropical Linear Spaces 53

3.1 The constant coefficient case . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1.1 Partition matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.2 Uniform matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.3 Fano plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Regular matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.1 Graphic matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.2 Cographic matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.3 The matroid R10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 Non-constant coefficient case . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4 Parametrizations of tropical linear spaces . . . . . . . . . . . . . . . . . . . 69

iii

4 Implicitization and Elimination 71

4.1 The tropical variety of a generic complete intersection . . . . . . . . . . . . 734.2 Tropical elimination and implicitization . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 From tropical variety to Newton polytope to implicit equation . . . . . . . . 944.4 TrIm: Software for tropical implicitization . . . . . . . . . . . . . . . . . . . 99

4.4.1 Algorithms and implementation . . . . . . . . . . . . . . . . . . . . . 1004.4.2 Computing tropical discriminants . . . . . . . . . . . . . . . . . . . . 106

Bibliography 111

iv

List of Figures

1.1 Tropical hyperplane in TP2 defined by tropical linear form a. . . . . . . . . 10

2.1 Grids representing x21x22x31x32x41x42 and x11x13x31x33x41x42 for r = 4, n =3. These are the labels of v1 and v2 in Figure 2.2. . . . . . . . . . . . . . . . 20

2.2 Tropical convex hull of four points in TP2. . . . . . . . . . . . . . . . . . . . 222.3 (a) Tropical tetrahedron in TP3 with black input vertices. The circles denote

“double vertices”. (b) After wiggling we can see all degenerate edges. . . . 272.4 (a) Tropical hyperplane in TP2 with apex a. (b) The sectors at apex 0 in

TP2. (c) Tropical halfspace (a, {1, 2}) in TP2. . . . . . . . . . . . . . . . . 282.5 (a) Tropical cyclic polytope on four vertices in TP3. (b) A path corresponding

to a generator of the Alexander dual I∗. . . . . . . . . . . . . . . . . . . . . 302.6 (a) Paths in grids corresponding to two one-valent vertices in Cr,n. (b) Hor-

izontal and vertical stripes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.7 (a) A diagonal step. (b) Corners indicating that the corresponding monomials

are not minimal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.8 The staircase diagram of the ideal in Example 2.19 . . . . . . . . . . . . . . 372.9 The three-dimensional part of this is the three cubes in the lower-left corner

of the figure; it also contains a pair of vertical two-dimensional flaps nearvertices C and F, and an extensive top flap connecting vertices A and B tothe other vertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.10 Various lifts of the tropical polytope in Figure 2.9. The faces in the back areunshaded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.11 The J-facet ABED, and two points which demonstrate that it is not anextreme set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.12 The face lattice of the model (L) and an artist’s rendition of it as a cellcomplex (R). The latter consists of a flat hexagon with a couple of squarespuckering up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.13 A tropical 3-polytope which is a more complicated version of the model. . . 482.14 A five-vertex tropical polytope in TP3: cube with pendant edge. . . . . . . 502.15 Four lifts of the cube with pendant vertex. Two are bipyramids and two are

square pyramids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

v

2.16 The face lattice of a cube with pendant edge (L) and its realization as a cellcomplex (R), a tetrahedron with one edge subdivided. . . . . . . . . . . . . 51

2.17 A tropical (2, 4)-hypersimplex, i.e. octahedron. . . . . . . . . . . . . . . . . 52

3.1 Fano plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 2-edge-connected graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 4-cycle-complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1 Tropical plane curves and their Newton polygons . . . . . . . . . . . . . . . 814.2 The Bergman complex of a transversal matroid . . . . . . . . . . . . . . . . 854.3 Tropical construction of a bicubic surface in 3-space . . . . . . . . . . . . . 90

vi

List of Tables

2.1 Monomial matrix M2 in Example 2.8. . . . . . . . . . . . . . . . . . . . . . 232.2 Computation times (in seconds) for I, I∗, and the free resolution for tropical

cyclic polytopes Cr,n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 The output of TrIm for the graph in Example 4.18 . . . . . . . . . . . . . . 1044.2 The tropical variety of the implicitization problem in Example 4.18. . . . . 1054.3 The output that describe the Newton polytope of the implicit equation and

the lattice points in it, for Example 4.18 . . . . . . . . . . . . . . . . . . . . 106

vii

Acknowledgments

First and formost, I am tremendously grateful to my advisor Professor Bernd Sturmfels for

providing guidance and encouragement starting even before my first day of graduate school

and for being a constant source of inspiration and optimism ever since. I thank Professors

Lior Pachter and Alper Atamturk for agreeing to be on my qualifying exam and dissertation

committees and providing comments on a draft of this thesis.

I am indebted to great mathematics teachers and professors who nurtured my in-

terest in mathematics over the years — my parents who were my first math teachers, Ms.

Susan Kelley at ALHS in San Francisco, and many inspiring professors at UC Davis. I

am heartened by how much they cared. In particular, I thank late Professor Evelyn Silvia

for teaching me how to write mathematics and Professor Jesus De Loera for much encour-

agement when I was an undergraduate student and continued belief in me as a graduate

student. I owe a great debt of gratitude to my undergraduate advisor Professor Motohico

Mulase, without whom I might not have pursued graduate studies in the first place, for

introducing me to mathematics research and for teaching me at least as much about mathe-

matics as mathematics. I am indebted to to wonderful professors at UC Berkeley for helping

me deepen my mathematical knowledge.

I thank my coauthors Florian Block, Mike Develin, Ingileif Hallgrımsdottir, Peter

Huggins, Michael Joswig, Alex Milowski, Jenia Tevelev, and Debbie Yuster, for sharing

moments of rapture and periods of frustration and for letting me use some of the joint works

in this thesis. I thank my fellow graduate students in the math department, especially my

awesome officemates, for making it a pleasure to go into Evans Hall every day. I also thank

viii

all the people I met at conferences, workshops, and summer schools, for just being there

and talking to me. Like my advisor always says, doing mathematics is a human activity,

and it has been a splendid experience for me.

I would like to acknowledge the National Science Foundation and the Graduate

Opportunity Program at UC Berkeley for funding my graduate education. I also thank the

Institute for Mathematics and Its Applications in Minneapolis for hosting my visit in Spring

2007. The postdocs, visitors, and staff at the IMA have made this semester a highlight of

my graduate studies.

Last but not least, I am immensely grateful to my family and friends for keeping

me sane, providing me invaluable moral support, and believing in me when it is difficult to

do so myself. This thesis would have been meaningless without their love.

1

Chapter 1

Introduction

In this chapter, we introduce the necessary concepts from polyhedral geometry,

and commutative algebra, and we present the basic definitions in tropical geometry.

1.1 Polyhedral geometry

Polyhedra are fundamental objects in geometric combinatorics whose combina-

torial properties and connections to commutative algebra have been intensively studied

[Zie95, MS05]. Since tropical objects are polyhedral complexes, we will first review basic

definitions of polyhedral geometry. This section is based on [Zie95].

A point set A in Rd is convex if for any two points a,b ∈ A, the line segment

{λa + (1 − λ)b : 0 ≤ λ ≤ 1} is also in A. For any set A ⊂ Rd, the convex hull conv(A)

is the smallest convex subset of Rd containing A, or the intersection of all convex subsets

containing A. Equivalently,

conv(A) = {λ1a1 + · · ·+ λkak : ai ∈ A,λi ≥ 0, λ1 + · · ·+ λk = 1}.

2

A polytope is the convex hull of a finite set of points.

A cone in Rd is a non-empty set that is closed under taking finite linear combi-

nations with non-negative coefficients. Every cone contains 0, and the only bounded cone

is the zero cone {0}. For any set A ⊂ Rd, we can define its conical hull or positive hull

cone(A) as the intersection of all cones containing A. Equivalently,

cone(A) = {λ1a1 + · · · + λkak : ai ∈ A,λi ≥ 0}.

We also denote this by R≥0A sometimes. This is the zero cone when A is {0} or empty.

A cone is called polyhedral if it is the conical hull of a finite set of points. All the cones

we encounter in this thesis are polyhedral, hence, we will use the word “cone” to mean

“polyhedral cone” unless otherwise noted.

The Minkowski sum of two sets P,Q ⊂ Rd is the set P+Q = {x+y : x ∈ P,y ∈ Q}.

A polyhedron is a set of the form conv(A) + cone(B) where A and B are finite subsets of

Rd. Note that a polytope is a bounded polyhedron. Familiar polytopes in low dimensions

include polygons, tetrahedra, cubes, and octahedra. The Minkowski sum of two polyhedra

is also a polyhedron.

A halfspace in Rd is a set of the form {x : c1x1 + · · ·+ cdxd ≥ 0}. A fundamental

theorem in polyhedral geometry states that the polyhedra are precisely intersections of

finitely many half spaces. The dimension of a polyhedron P is the dimension its affine hull

{λ1a1 + · · · + λkak : ai ∈ P, λ1 + · · ·λk = 1}. A d-dimensional polyhedron in Rd is called

full-dimensional.

A face of a polyhedron P is the intersection of P with the boundary of a halfspace

containing P . Faces of polyhedra are again polyhedra. The empty set and P itself are

3

always faces of P . The other faces are called proper faces. The smallest and largest proper

faces P are called the vertices and facets of P , respectively. A polytope is the convex hull

of its vertices and, if it is full-dimensional, the intersection of the facet-defining halfspaces.

A k-dimensional polytope is called simple if every vertex is in exactly k facets, and it is

called simplicial if every facet contains exactly k vertices. A k-dimensional polytope with

exactly k + 1 vertices or k + 1 facets is called a k-simplex. A triangle is a 2-simplex, and a

tetrahedron is a 3-simplex. The face lattice of a polytope is the partially ordered set (poset)

of all faces, ordered by inclusion.

A polyhedral complex C in Rd is a finite collection of polyhedra in Rd such that

1. the empty polytope is in C,

2. if a polyhedron is in C, then all of its faces are also in C,

3. the intersection of any two polyhedra in C is a face of both.

The dimension of C is the largest dimension of polyhedra in it. A maximal face of C is a

polyhedron in C that is not contained in any other polyhedra in C. A polyhedral complex

is pure if all of its maximal faces have the same dimension. The ground set C is the union

of polyhedra in it. A polytopal complex is a polyhedral complex in which all polyhedra are

bounded.

A fan F in Rd is a finite collection of non-empty polyhedral cones in Rd such that

1. if a cone is in F , then all of its non-empty faces are also in F ,

2. the intersection of any two cones in F is a face of both.

4

The lineality space of a fan F is the maximal linear space that is contained in all the cones

in F . A fan F is called complete if the union of cones in F is all of Rd. The common

refinement of two complete fans F and G is the fan {C ∩ C ′ : C ∈ F , C ′ ∈ G}.

The (inner) normal fan of a non-empty polytope in Rd is the complete fan consist-

ing of the cones which are all linear functions minimal on a fixed face of the polytope. The

normal fan of a Minkowski sum of two polytopes is the common refinement of the normal

fans of the summands.

A (polytopal) subdivision of a polytope P is a polytopal complex whose ground

set is P . A subdivision is called a triangulation if all the polytopes in the complex are

simplices. Let P be a polytope in Rd with n vertices v1, . . . ,vn. Every vector w ∈ Rn

induces a subdivision of P as follows. Let Q be the polyhedron

conv{(v1, w1), . . . , (vn, wn)}+ cone(0, . . . , 0, 1) ⊂ Rd+1.

The induced subdivision is the polytopal complex consisting of the projections of the

bounded faces of Q onto the first d coordinates. The subdivisions that arise this way

are called regular.

If the polytope P is full-dimensional, the dual F of the regular subdivision is

a polyhedral complex defined as followed. To each bounded facet of Q supported by a

hyperplane c · x ≥ 1, associate the point c ∈ Rd. For each other lower face of Q, the dual

is the convex hull of points corresponding to the facets of Q containing that face. The dual

complex consists of those dual faces.

5

1.2 Commutative algebra

In this section, we will define some basic concepts from commutative algebra.

Much of the material is based on [MS05].

Let R = k[x] = k[x1, . . . , xn] be the ring consisting of polynomials in variables

x1, . . . , xn with coefficients in the field k. An ideal I ⊂ R is a subset that is closed under

addition and multiplication by elements in R.

A monomial in R is a product xa = xa11 · · · x

ann where a = (a1, . . . , an) ∈ Zn is a

vector of nonnegative integers. A monomial is called squarefree if a ∈ {0, 1}n. An ideal

generated by monomials is called a monomial ideal.

Let w ∈ Rn. Let the weight of xi be wi and the weight of a monomial xa =

∏

xai

i ∈ R be the dot product∑

aiwi. For a polynomial f =∑

cixai , we define the initial

form inw(f) to be the sum of terms cixai such that xai has maximal weight. A polynomial

f is called homogeneous with respect to a weight w if f = inw(f). An ideal is homogeneous

with respect to w if it is generated by a set of homogeneous polynomials. The initial ideal

of an ideal I ⊂ R is inw(I) = 〈inw(f) : f ∈ I〉. For most choices of w, the initial ideal is

a monomial ideal. A Grobner basis I with respect to w is a subset {f1, . . . , fr} ⊂ I such

that inw(I) = 〈inw(f1), . . . , inw(fr)〉. If the ideal I is homogeneous with respect to some

positive weight vector w ∈ Rn>0, then we can partition Rn such that all weight vectors w in

the same part give the same initial ideal of I. This partition forms a complete polyhedral

fan called the Grobner fan G of I.

We will now review some notions of monomial ideals. We will denote by R(−a)

the free R-module generated by an element in degree a. An R-module M is Zn-graded

6

if M = ⊕b∈ZnMb, and xaMb ⊆ Ma+b. A monomial ideal is a Zn-graded R-module. A

free R-module of finite rank is a direct sum F ∼= Rr of copies of R, for some nonnegative

integer r. For us, F will be Zn-graded, i.e. F ∼= R(−a1) ⊕ · · · ⊕ R(−ar) for some vectors

a1, . . . ,ar ∈ Zn. A free resolution of an R-module M is a sequence

F· : 0∂−1←−M

∂0←− F0∂1←− F1 ←− · · · ←− Fl−1

∂l←− Fl∂l+1←− 0

of maps where each Fi is a free R-module and ker(∂i−1) = im(∂i) for all 0 ≤ l + 1. Here we

assume that each map ∂i is degree-preserving, i.e. the image of a degree-m element is again

degree-m.

In [BS98], Bayer and Sturmfels defined the hull complex, a complex which yields

a resolution of a monomial ideal I ⊂ R. The construction is as follows: lift each generator

xa ∈ k[x1, . . . , xn] to a vector ta ⊂ Rd−1, where t is some large real number. Form a

polyhedron Pt by adding the positive orthant to the convex hull of these vectors. For t

sufficiently large, it was shown that this polyhedron has constant combinatorial type, and

that the complex of bounded faces of Pt yields a cellular resolution of the monomial ideal.

We give a brief review of the process which leads from a complex to a resolution.

Given any polytopal complex P with the vertices labeled by generators xa, we label

each face by the least common multiple of the generators corresponding to its vertices. We

then form a chain complex XP as follows: each face F of the polytopal complex corresponds

to a generator lying in homological degree equal to its dimension. This chain complex is to

be graded, with the degree of F equal to the label of F (which we denote xF ); each generator

maps to the appropriately homogenized signed sum of the generators corresponding to its

7

facets, i.e.

∂F =∑

±xF

xF ′

F ′, (1.1)

where the sum runs over all facets F ′ of F and the signs are chosen so that ∂2 = 0. The

complex XP is a cellular resolution if it is acyclic, i.e. has homology only in degree 0.

Given any degree b ∈ Zn, the complex X�b is defined to be the subcomplex of this

chain complex given by taking the generators with xF dividing xb. The key result about

cellular resolutions is the following:

Proposition 1.1 ([BS98]). The cellular free (algebraic) complex supported on a labelled

cell complex X is a cellular resolution if and only if X�b is acyclic over the ground field k

for all vectors b with non-negative integer entries. When the complex is acyclic, it is a free

resolution of S/I where I is the ideal generated by the monomial labels of the vertices.

1.3 Tropical mathematics

The tropical semiring T = (R ∪ {∞},⊕,⊙) is the set R of real numbers with two

binary operations called tropical addition ⊕ and tropical multiplication ⊙ defined as

a⊕ b = min(a, b), and a⊙ b = a + b, for all a, b ∈ R.

Then Rn has the structure of a semimodule over the semiring (R,⊕,⊙) with tropical addition

(x1, . . . , xn)⊕ (y1, . . . , yn) = (x1 ⊕ y1, . . . , xn ⊕ yn),

and tropical scalar multiplication

c⊙ (x1, . . . , xn) = (c⊙ x1, . . . , c⊙ xn).

8

Just as ordinary linear and polynomial algebra give rise to polyhedral and algebraic geom-

etry, the tropical linear and polynomial algebra give rise to tropical convex geometry and

tropical algebraic geometry [Stu02, SS04, Mik06, DS04, Jos05].

The term “tropical semiring” was coined by Dominique Perrin and Jean-Eric Pin

in the honor of their Brazilian colleague Imre Simon, a developer of the “min-plus algebra”

[Pin98]. The tropical (min-plus) arithmetics has had successful applications in computer

science and optimization, and recently tropical geometry has found new applications in

commutative algebra and algebraic geometry.

Tropical geometry is a piecewise linear shadow of algebraic geometry. A way in

which tropical objects arise naturally is as images of objects in Kd, a vector space over the

field of generalized Puiseux series, with respect to its non-Archimedean valuation. For a

field k, let K = k((tR)) be the set of formal power series of the form f =∑

a∈R cata where

supp(f) = {a ∈ R : ca 6= 0} is artinian, i.e. every strictly decreasing sequence of elements

in it is finite. Then K forms a field with valuation deg : f 7→ min(supp(f)) onto R. By

[Rib92], K is algebraically closed (resp. real closed) if k is algebraically closed (resp. real

closed). Note that deg(f · g) = deg(f)⊙ deg(g) and deg(f + g) ≥ deg(f)⊕ deg(g).

1.3.1 Tropical convex geometry

Tropical convexity was introduced by Develin and Sturmfels in [DS04] and has

motivated several research projects in this direction [Jos05, BY06, DY07]. A set A ⊂ Rn is

called tropically convex if for all x,y ∈ A and a, b ∈ R also (a⊙x)⊕ (b⊙y) ∈ A. Notice that

we do not put any extra condition on a and b as in usual convexity. The tropical convex

hull tconv(V ) of a set V ⊂ Rn is the minimal tropically convex set containing V in Rn with

9

respect to inclusion. Also,

tconv(V ) = {(a1 ⊙ v1)⊕ · · · ⊕ (ar ⊙ vr) : v1, . . . ,vr ∈ V and a1, . . . , ar ∈ R}.

A tropical polytope is the tropical convex hull of a finite set of points. Since any tropically

convex set A is closed under tropical scalar multiplication, we identify it with its image

under the projection onto the (n− 1)−dimensional tropical projective space

TPn−1 = Rn/(1, . . . , 1)R.

An important structure of tropical polytopes is the following:

Theorem 1.2 ([DS04]). Tropical polytopes are usual polyhedral complexes dual to regular

subdivisions of products of simplices ∆r−1 ×∆n−1.

Although the definition of tropical polytopes looks like a definition of a “tropical

linear span”, these objects do behave like polytopes. They are bounded in the tropical pro-

jective space. In [DS04], Develin and Sturmfels showed that they satisfy tropical analogues

of the Caratheodory’s theorem and the Farkas Lemma.

Let K = R((tR)) be the field of Puiseux with real coefficients as defined above.

We will see in Chapter 2 that tropical polytopes are precisely the images of polytopes in

(K>0)n under the non-archimedean valuation.

In Chapter 2, we will prove two results that relate tropical polytopes and cellular

resolutions, one using the polyhedral complex structure of tropical polytopes, and another

using the combinatorial structure of their lifts to the field of Puiseux series. This chapter

is based on [BY06] and [DY07].

10

1.3.2 Tropical linear spaces

For a tropical linear form (a1⊙ x1)⊕ · · · ⊕ (an⊙ xn) where ai ∈ R∪ {−∞}, define

its tropical hyperplane to be the set of (x1, . . . , xn) ∈ Rn such that the minimum value in

the tropical linear form is attained at least twice. See figure 1.1. Recall that the tropical

addition is taking minimum.

(1, 0, 0) = (0,−1,−1)

(0, 1, 0)

(0, 0, 1)

−a

Figure 1.1: Tropical hyperplane in TP2 defined by tropical linear form a.

In an ordinary vector space, linear spaces can be defined either as intersections

of hyperplanes or linear spans of some vectors. However, in the tropical world, not every

intersection of tropical hyperplanes or tropical linear span behaves well. In fact, arbitrary

tropical linear spans give tropical polytopes as seen before. However, for some special sets

of hyperplanes, called tropical bases, and for some special sets of vectors called cocircuits,

the intersections and tropical linear spans respectively give tropical linear spaces. This is

the topic of study in Chapter 3. We will explore the tropical bases of some families of

tropical linear spaces. We will also show that a tropical linear space is the “tropical linear

span” or the tropical convex hull of a special set of vectors at infinity.

The constant coefficient tropical linear spaces are also called Bergman fans of

11

matroids [AK06] and they play a crucial role in the theory of tropical discriminants [DFS].

1.3.3 Tropical algebraic geometry

While working with tropical varieties, we will adopt the convention in [SS04,

STY06] and define the initial form of a polynomial to be the sum of its minimal degree

terms instead of the more commonly used maximal degree. Otherwise, we will be working

in the max-tropical semiring. All the theory, of course, is the same.

Let I ⊂ C[x] be an ideal in the polynomial ring over complex numbers. The tropical

variety T (I) of the ideal I is defined to be the set of w ∈ Rn such that the initial ideal inw(I)

does not contain a monomial [SS04, BJS+]. If I is homogeneous, T (I) is naturally a subfan of

(the negative of) the Grobner fan of I. Otherwise, one can put a polyhedral fan structure on

T (I), for example, the Grobner fan structure of a homogenization of I. If I is prime, T (I) is

pure, has dimension the same as the Krull dimension of I [BG84, Stu02], and is connected

in codiimension 1 [BJS+]. They also satisfy tropical analogues of Bezout and Bernstein

theorems [Stu02]. Since they are polyhedral objects, the invariants of the original varieties

such as the dimesion and the degree can be found using polyhedral computations. Tropical

varieties are also known under the names logrithmic limit sets [Ber71], non-archimedean

amoebas [EKL], and Bieri–Groves sets [BG84].

For a polynomial f =∑

caxa ∈ C[x], the Newton polytope P of f is the convex

hull of points a ∈ Rn such that ca 6= 0. The tropical hypersurface T (f) of f is the union

of codim-1 cones in the inner normal fan of the Newton polytope P . The tropical variety

T (I) is the intersection of tropical hypersurfaces T (f) for all f ∈ I. The Newton polytope

of a product of two polynomials f and g is the Minkowski sum of the Newton polytopes of

12

f and g. Recall that the normal fan of a Minkowski sum is the common refinement of the

summands, so T (fg) = T (f) ∪ T (g). If I is the principal ideal generated by a polynomial

f , then T (I) = T (f).

Every maximal cone Γ of the fan T (I) naturally comes with a multiplicity, which is

a positive integer. The multiplicity MΓ of a d-dimensional cone Γ is the sum of multiplicities

of all monomial-free minimal associate primes of the initial ideal inv(I) in C[x1, . . . , xn]

where v ∈ Rn is any point in the relative interior of the cone Γ. The multiplicities MΓ

are an important piece of data which must also be determined when computing a tropical

variety T (I). In the hypersurface case, a maximal cone Γ of T (f) correspond to an edge

E of the Newton polytope of f , and the multiplicity of Γ is the lattice length of E, i.e. one

less than the number of integer points in E. It is with these multiplicities that a tropical

variety satisfies the balancing condition which ensures that tropical intersection numbers

are independent of choices; see [Mik06, §3] and [Kat, §6.4].

In Chapter 4, we will present a tropical approach to the implicitization problem,

which is the problem of finding the defining ideal of a parametrized variety. We give a

combinatorial construction of the corresponding tropical variety, from which one can obtain

information about the original variety, such as the Chow polytope. In the important special

case when the parametrized variety is a hypersurface, this gives a description of the Newton

polytope of the implicit equation in terms of the Newton polytopes of the polynomials that

define the parametrization, hence answering a decade-old problem posted in [SY94]. We will

also describe a computer implementation of this method. This chapter is based on [STY06].

13

Chapter 2

Tropical Polytopes and Cellular

Resolutions

Recall that a set A ⊂ Rn is called tropically convex if for all x, y ∈ A and a, b ∈ R

we have (a⊙x)⊕(b⊙y) ∈ A, where the tropical addition and scalar multiplications are taken

coordinatewise. The tropical convex hull tconv(V ) of a set V ⊂ Rn is the inclusion-wise

minimal, tropically convex set containing V in Rn. The tropical convex hulls of finite sets

of points are called tropical polytopes. In [DS], Develin and Sturmfels showed that tropical

convex hulls have a natural polyhedral cell structure which is polar dual to subdivisions of

products of simplices.

In Section 2.1, we will exploit this cellular structure to realize the tropical polytopes

as minimal cellular resolutions of some particular monomial ideals. We will use this result

to give a practical algorithm for computing tropical polytopes in Section 2.1.1 and describe

an implementation using Macaulay 2 [GS02].

14

Tropical polytopes arise naturally as images of polytopes under a non-Archimedean

valuation. This new point of view gives rise to a construction of a family of cellular res-

olutions of arbitrary monomial ideals, which generalizes the hull resolution by Bayer and

Sturmfels [BS]. This is the topic of Section 2.2

Although Develin and Sturmfels gave a polyhedral cell structure of tropical poly-

topes in [DS04], this description is unsatisfactory in many ways as a definition of the face

poset of tropical polytopes. For instance, the tropical convex hull of three points can have

as many as six vertices. A more natural approach is to define faces of tropical polytopes as

the “shadows” of faces of ordinary polytopes over the Puiseux series field. A new definition

will be proposed in Section 2.2.1. This chapter is based on joint works with Florian Block

[BY06] and Mike Develin [DY07].

2.1 Tropical polytopes via cellular resolutions

Let V = {v1, . . . ,vr} ⊂ TPn−1, vi = (vi1, . . . , vin), and P = tconv(V ). We first

describe a natural polyhedral complex structure of the tropical polytope P. Define an

unbounded polyhedron as follows:

PV = {(y, z) ∈ Rr+n/(1, . . . , 1,−1, . . . ,−1)R : yi + zj ≤ vij for all i ∈ [r], j ∈ [n]}.

By [DS04], there is a piecewise linear isomorphism between the complex of bounded faces

of PV and the tropical polytope P = tconv(V ) given by the projection (y, z) 7→ z. The

boundary complex ∂PV of PV is polar to the regular polyhedral subdivision of the product

of simplices ∆r−1 × ∆n−1 induced by the weights vij. We denote this regular subdivision

by (∂PV )∗. More precisely, a subset of vertices (ei, e′j) of ∆r−1 ×∆n−1 forms a cell of the

15

subdivision (∂PV )∗ if and only if the equations yi+zj = vij indexed by these vertices specify

a face of the polyhedron PV .

Let S = R[x11, . . . , xrn] be the polynomial ring with indeterminates xij for i ∈

[r] = {1, . . . , r} and j ∈ [n] = {1, . . . , n}. Let the weight of xij be vij .

Let A denote the (r+n)×rn integer matrix whose column vectors are the vertices

(ei, e′j) of ∆r−1 ×∆n−1, where i ∈ [r], j ∈ [n]. This defines a homomorphism Zrn → Zr+n

by eij 7→ (ei, e′j). Let L denote its kernel. The lattice ideal

J = 〈xa − xb : a,b ∈ Nrn with a− b ∈ L〉.

of ∆r−1 × ∆n−1 is generated by the 2 × 2 minors of the r × n matrix of indeterminates

[xij ]. Let I = inV (J) be the initial ideal of J with respect to weight V . Since the ideal

J is homogeneous with respect to any grading assigning the same weight to the variables

in each row, the initial ideal I is independent of the representatives of the points vi in the

tropical projective space TPn−1. In other words, if c · (1, 1, . . . , 1) is added to any vi, the

initial ideal I remains the same.

In the rest of this section, we will assume that the points v1, . . . ,vr are in generic

position, i.e., they satisfy the following equivalent conditions.

Proposition 2.1. The following are equivalent.

(1) The initial ideal I is a monomial ideal.

(2) The regular subdivision (∂PV )∗ of ∆r−1 ×∆n−1 induced by the weights vij is a trian-

gulation.

(3) The polyhedron PV is simple.

16

(4) For any k distinct points in V , their projections onto a k-dimensional coordinate

subspace do not lie in a tropical hyperplane, for any 2 ≤ k ≤ n.

(5) No k × k submatrix of the r × n matrix [vij ] is tropically singular, i.e., has vanishing

tropical determinant (e.g. see [DS04]), for any 2 ≤ k ≤ n.

Proof. (2) ⇐⇒ (3) follows directly from the polarity between the regular subdivisions of

∆r−1 ×∆n−1 and ∂PV .

(2) ⇐⇒ (5) is proven in [DS04, Proposition 24].

(4) ⇐⇒ (5) is proven in [RGST05, Lemma 5.1].

(1) ⇐⇒ (2): Statement (1) is equivalent to V being in the interior of a full dimensional

cone in the Grobner fan of the lattice ideal J . Statement (2) means that V is in the interior

of a full dimenional cone in the secondary fan N (Σ(A)) which is the normal fan of the

secondary polytope of A (for details see [Stu96]). By [Stu96, Proposition 8.15(a)], these

two fans coincide if A is unimodular, i.e., all invertible rank(A) × rank(A) submatrices

have the same determinant up to sign. We will check criterion (iv) of [Sch86, Theorem 19.3]

for total unimodularity. Fix a collection of rows of A. Split it according to containment in

the upper r× rn submatrix of the (r +n)× rn matrix A. Then the sum of the rows in each

part is a 0-1 vector. This implies that all submatrices of A have determinants 0 or ±1, so

A is unimodular.

It also follows from the unimodularity that all monomial initial ideals of J are

square free [Stu96, Corollary 8.9]. The square free Alexander dual I∗ of a square free

monomial ideal I = 〈xa1, . . . ,xak

〉 is

I∗ = ma1∩ · · · ∩ma

k

,

17

where each ai is a 0-1 vector and ma = 〈xj : aj = 1〉. See [MS05, Stu96] for details. The

following is the main result of this section.

Theorem 2.2. For a sufficiently generic set of points V in TPn−1, the tropical convex hull

P = tconv(V ) supports a minimal linear free resolution of the ideal I∗, as a cellular complex.

Moreover, the cellular structure of the minimal free resolution is unique.

Since the set of 2×2 minors of a matrix is fixed under transposition of the matrix,

we immediately see the duality between tropical convex hulls of r points in TPn−1 and n

points in TPr−1, as shown in [DS04].

Let ∆V (J) be the initial complex of J , i.e. a simplicial complex whose faces are

supports of monomials that are not in I = inV (J). In other words, the Stanley-Reisner

ideal of ∆V (J) is I = inV (J). We can identify a square free monomial m ∈ S with the

set of indeterminates xij dividing m. The vertices of ∆V (J) are xij , and the minimal

generators of I are the minimal non-faces of ∆V (J). Moreover, the minimal generators of

the Alexander dual I∗ are the complements of the maximal cells of ∆V (J). The following

lemma establishes a connection between the ideal J and the tropical convex hull.

Lemma 2.3. The cell complexes ∆V (J) and (∂PV )∗ are isomorphic. In particular, there is

a bijection between maximal cells of ∆V (J) and those of (∂PV )∗ induced by xij ←→ (ei, e′j).

Proof. The result follows from applying [MS05, Theorem 7.33] or [Stu96, Theorem 8.3] to

the product of simplices ∆r−1 ×∆n−1.

We will label the vertices of PV by the minimal generators of I∗ so that PV gives

a cellular resolution of I∗. First, we have a general lemma about simple polyhedra and

cellular resolutions.

18

Lemma 2.4. Let P be a simple polyhedron (possibly unbounded) with facets F1, . . . , Fm. La-

bel each face G of P by xaG =∏

Fi+G xi ∈ R[x1, . . . , xm]. Then the complex of bounded faces

of P supports a minimal linear free resolution of the square free monomial ideal generated

by the vertex labels.

Proof. Let X be the cellular free complex supported on the bounded faces of P . To see that

the complex is a free resolution, by Proposition 1.1, it suffices to check that each subcomplex

X�b is acyclic for all b ∈ Nm. Since the monomial labels are squarefree, it suffices to check

for b ∈ {0, 1}m. Notice that a face G is in X�b if and only if xaG =∏

Fi+G xi divides xb if

and only if G ⊂⋂

i:bi=0 Fi. Since⋂

i:bi=0 Fi is contractible, X�b is acyclic over any ground

field. Hence X is a free resolution. Moreover, every face of P has a unique label since a

face is uniquely determined by the set of facets containing it. So the resolution is minimal.

Since the polytope P is simple, any codimension-k face is contained in k facets, and its

label has degree m− k, so the resolution is linear.

We will apply this to PV to prove Theorem 2.2.

Proof of Theorem 2.2. Since V is generic, PV is simple. Hence, by Lemma 2.4, the tropical

convex hull P, which is isomorphic to the complex of bounded faces of PV , supports a

minimal linear free resolution of the ideal generated by the monomial labels of its vertices.

We will show that the labels from Lemma 2.4 coincide with the minimal generators of I∗.

The facets Fij of PV are defined by equations yi + zj = vij . Let xij be the

19

indeterminate corresponding to Fij . For a square free monomial m,

m is a vertex label of PV

⇐⇒⋂

{Fij : xij does not divide m} is a vertex of PV

⇐⇒ {(ei, ej) : xij does not divide m} is a maximal cell of (∂PV )∗

⇐⇒ {xij : xij does not divide m} is a maximal cell of ∆V (J)

⇐⇒ m is a minimal generator of I∗.

The third equivalence follows from Lemma 2.3.

By construction, the monomial labels are unique, so all the multi-graded Betti

numbers are at most one. The uniqueness of the cellular structure follows from the following

lemma.

Lemma 2.5. The monomial matrix Mi is unique up to rescaling the columns by non-zero

scalars in k if and only if βi,a ≤ 1 for all a and the monomials in Ai = {xa : βi,a 6= 0} do

not divide each other.

Proof.

⇐: Let a ∈ Nn be such that βi,a = 1 and no other monomial in Ai divides xa. Then

the a-graded piece of the ith syzygy module, which is the image of Mi in Fi−1, is a 1-

dimensional k-vector space. There is a unique linear transformation, up to rescaling, be-

tween two 1-dimensional vector spaces, so the column in Mi corresponding to S(−a) is

unique up to rescaling.

⇒: Suppose there are a,b ∈ Ai such that xa divides xb. Then the b-graded piece of the

ith syzygy module is at least 2-dimensional. In the monomial matrix Mi, we can add the

a-column to the b-column without changing its image. By the minimality of the resolution,

20

these columns are linearly independent, so this operation is not rescaling, i.e., we get a

different monomial matrix.

For the case βi,a > 1, take a = b in the previous argument.

However, the multi-graded Betti numbers already determine the tropical polytope

because in this case a face F contains a face G if and only if the monomial label of F is

divisible by the monomial label of G. Moreover, the vertex labels (the minimal generators

of I∗) determine all the other monomial labels by Lemma 2.12.

The dimension dim(U) of any subset U of TPn−1 is the affine dimension of its

projection {u ∈ Rn−1 : (0, u) ∈ U} onto the last n− 1 coordinates.

Corollary 2.6. For any face F ⊂ P, dim(F ) = deg(xaF )− (n− 1)(r − 1).

Proof. Since the simple polyhedron PV has dimension r + n − 1 and rn inequalities, by

construction the degree of a vertex label is rn − (r + n − 1) = (r − 1)(n − 1). Then the

assertion follows from the fact that the resolution is linear.

Figure 2.1: Grids representing x21x22x31x32x41x42 and x11x13x31x33x41x42 for r = 4, n = 3.These are the labels of v1 and v2 in Figure 2.2.

The monomial labels have a geometric meaning. To have a more intuitive notation,

we will represent each squarefree monomial m ∈ S with an r×n grid shaded at position (i, j)

if xij does not divide m. Hence the support of xaF is left unshaded in the grid (see Figure

2.1). Let Cj = cone{ei : i 6= j} be the closed cone which is the usual conical (positive) hull

21

of all but one standard unit vector. Suppose z = (z1, . . . , zn) ∈ P is in the relative interior

of a cell with label xaz , and it is the image of the point (y, z) ∈ PV . Then

xij 6 | xaz ⇐⇒

yi + zj = vij

yi + zk ≤ vik ∀k

⇐⇒ vij − zj ≤ vik − zk ∀k

⇐⇒ vi − z ∈ Cj ⇐⇒ vi ∈ z + Cj.

So the box (i, j) is shaded if and only if the input vertex vi lies in the sector z + Cj . See

Figure 2.4(b).

This monomial labeling is essentially the same as the labeling by types introduced

in [DS04]. Specifically, for any point z in the relative interior of a cell F in P with type(z) =

(S1, . . . , Sn), we have i ∈ Sj if and only if xij does not divide xaF . The following result

follows from [DS04, Lemma 10].

Lemma 2.7. Given the monomial label xaz of a vertex z, its coordinates can be computed

by solving the linear system

{zl − zk = vil − vik : i ∈ [r], k, l ∈ [n], xik and xil do not divide xaz}.

Example 2.8 (Four Points in TP2). Suppose we are given the following points in TP2

(r = 4, n = 3):

v1 = (0, 3, 6),v2 = (0, 5, 2),v3 = (0, 0, 1),v4 = (0, 4,−1).

They determine the tropical polytope in Figure 2.2.

The points give the weight vector V = (0, 3, 6, 0, 5, 2, 0, 0, 1, 0, 4,−1) in the poly-

nomial ring

S = R[x11, x12, x13, x21, x22, x23, x31, x32, x33, x41, x42, x43].

22

v = (0,0,1)

v = (0,5,2)2

1v = (0,3,6)

f1

2

f3

e

v = (0,4,−1)4

3

f

Figure 2.2: Tropical convex hull of four points in TP2.

The initial ideal I and its Alexander dual are

I = inV

⟨

2× 2 minors of

x11 x12 x13

x21 x22 x23

x31 x32 x33

x41 x42 x43

⟩

= 〈x33x41, x23x41, x23x31, x12x31, x31x11x42, x11x42, x31x41, x31x31, x13x21,

x33x42, x22x41, x13x32, x22x31, x11x22, x23x42, x22x33, x13x42, x13x22, x12x21x33〉,

I∗ = 〈x12x13x22x23x33x42, x12x13x22x23x41x42, x13x22x23x31x33x42,

x12x13x22x31x41x42, x13x22x31x33x41x42, x13x21x22x31x41x42,

x11x13x22x23x31x33, x21x22x31x32x41x42, x11x13x31x33x41x42,

x11x13x23x31x33x41〉.

Note that I is not generated in degree 2. Compare the minimal generators of I∗ with the

23

grids in Figure 2.2. The minimal free resolution of I∗ is of the form

0←− I∗M0←− S10 M1←− S12 M2

←− S3 ←− 0.

The tropical convex hull consists of 10 zero-dimensional faces (vertices), 12 one-dimensional

faces (edges), and 3 two-dimensional faces.

Table 2.1 shows the monomial matrix M2 in the monomial matrix notation of

[MS05, Section 1.4]. The rows correspond to the edges of the tropical polytope, and the three

columns, whose labels are omitted here, correspond to the faces f1, f2, and f3, respectively.

x12x13x22x23x31x33x42

x12x13x22x23x33x41x42

x12x13x22x23x31x41x42

x12x13x22x31x33x41x42

x11x13x22x23x31x33x42

x12x13x21x22x31x41x42

x13x22x23x31x33x41x42

x13x21x22x31x32x41x42

x11x13x22x31x33x41x42

x13x21x22x31x33x41x42

x11x13x22x23x31x33x41

x11x13x23x31x33x41x42

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

1 0 0

−1 0 0

−1 0 0

−1 1 0

0 0 −1

0 −1 0

1 0 −1

0 0 0

0 0 1

0 1 0

0 0 −1

0 0 1

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

Table 2.1: Monomial matrix M2 in Example 2.8.

2.1.1 An algorithm for computing tropical convex hulls

We will now discuss an algorithm and implementation for computing tropical poly-

topes. This algorithm has immediate applications to tropical linear spaces that will be

studied in Chapter 3.

24

Since the cellular structure of the minimal free resolution is unique, we get the

following algorithm for computing the tropical convex hull of a finite set of points in tropical

projective space.

Algorithm 2.9.

Input A list of points v1, . . . ,vr ∈ TPn−1 in generic position.

Output The tropical convex hull of the input points.

Algorithm

1. Set J = 〈2× 2 minors of the r × n matrix [xij]〉.

2. Compute I = inV (J).

3. Compute the Alexander dual I∗ of I.

4. Find a minimal free resolution of I∗.

5. Output the desired data about the tropical polytope.

Let 0 ← I∗ ← F0 ← · · · ← Fm be the free resolution computed by the algorithm,

and let Mi : Fi → Fi−1 denote the monomial matrices defining the boundary maps. Since

the free resolution is linear, the row labels of the matrix Mi are in one-to-one correspondence

with the faces of dimension i− 1, its column labels with the faces of dimension i. An entry

in Mi is nonzero if and only if its row label divides its column label, which happens if and

only if the face corresponding to its column contains the face corrresponding to its row.

Therefore the number of i-dimensional faces with k facets in the tropical convex hull is

equal to the number of columns of Mi having k nonzero entries. A face F is maximal if

25

and only if either it has dimension n − 1 or the row in Mdim(F )+1 labeled by xaF contains

zeroes only.

We can also compute the f -matrix [fij ] (0 ≤ i ≤ n − 1, 1 ≤ j) where fij is the

number of faces having dimension i and j vertices. We already know the f -vector∑

j fij

which is the sum of columns in the f -matrix. The following result in [DS04] was obtained

by counting regular triangulations of ∆r−1 ×∆n−1.

Proposition 2.10 ([DS04, Corollary 25]). All tropical convex hulls of r generic points

in TPn−1 have the same f -vector. The number of faces of dimension i is equal to the

multinomial coefficient

(

r + n− i− 2

r − i− 1, n− i− 1, i

)

=(r + n− i− 2)!

(r − i− 1)! · (n− i− 1)! · i!.

Example 2.11 (Four Points in TP2 - continuation of Example 2.8). The eighth row of

M1 corresponds to the vertex v1 (see Figure 2.2): It contains only one nonzero entry, so

the vertex v1 is contained in only one edge. The first column of M2 represents the two-

dimensional face f1 having five facets. This is reflected in five nonzero entries in that column.

The face f2 with three facets correspond to the second column which has 3 nonzero entries,

and the last column of M2 to f3. The eighth row of M2 is a zero row. This shows that the

one-dimensional face e is not contained in any other face, so it is maximal. The f−matrix

of this polytope is

10 0 0 0 0 0 · · ·

0 12 0 0 0 0 · · ·

0 0 1 0 2 0 · · ·

.

26

An alternative to free resolutions

The multi-graded Betti numbers already determine the tropical polytope because

in this case a face F contains a face G if and only if the monomial label of F is divisible

by the monomial label of G. Moreover, the vertex labels (the minimal generators of I∗)

determine all the other monomial labels by the following result, which follows from [DS04,

Corollary 14] and Corollary 2.6.

Lemma 2.12. Let F be a face of P with grid aF and let b be a grid arising from aF by

unshading one box such that no row or column is completely unshaded. Then there is a face

G ⊃ F with label aG = b and of one dimenstion higher.

Conversely, every face can be obtained this way starting from the vertices. So,

instead of algebraically computing the free resolution, we can build the fact poset com-

binatorially if we know the vertex labels, i.e., the minimal generators of I∗. We have an

implementation of this method using Macaulay 2 [GS02], Maple, and JavaView.

Non-generic input vertices

When the input vertices V are not in generic position, the initial ideal I is not

monomial. In that case, we can replace the weights V with any refinement which makes I

a monomial ideal and proceed as before to build the face poset. We can then compute the

coordinates of the vertices using Lemma 2.7 and identify vertices with the same coordinates.

Example 2.13. We can compute the following tropical tetrahedron in TP3 :

tconv{−e1,−e2,−e3,−e4}

= tconv{(0, 1, 1, 1), (0,−1, 0, 0), (0, 0,−1, 0), (0, 0, 0,−1)}.

27

These input vertices are not in generic position, so we end up with “double vertices” which

have the same coordinates (see Figure 2.3(a)). The degenerate edges can be seen when we

wiggle the input vertices to (0, 1, 1, 1), (0,−1,−0.1, 0), (0, 0,−1,−0.1) and (0,−0.1, 0,−1).

The “wiggled tropical tetrahedron” is shown in Figure 2.3(b). It is an example of a generic

tropical polytope in TP3 which is also a usual polytope.

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(a) (b)

Figure 2.3: (a) Tropical tetrahedron in TP3 with black input vertices. The circles denote“double vertices”. (b) After wiggling we can see all degenerate edges.

Tropical halfspaces

Tropical halfspaces introduced in [Jos05] give us an exterior description of tropical

polytopes. We can extend our algorithm to find such a description.

The tropical hyperplane at the apex a ∈ TPn−1 is the set which is the union

of boundaries of the sectors a + Ci (see Figure 2.4). For a ∈ TPn−1, ∅ 6= A ( [n], the

set a +⋃

i∈A Ci is a closed tropical halfspace (see Figure 2.4(c)). Tropical halfspaces are

tropically convex, and a tropical polytope P is the intersection of the inclusionwise minimal

halfspaces containing it [Jos05]. The apex of such a minimal halfspace must be a vertex

28

−b2

C3

C1C0 a

(a) (b) (c)

Figure 2.4: (a) Tropical hyperplane in TP2 with apex a. (b) The sectors at apex 0 in TP2.(c) Tropical halfspace (a, {1, 2}) in TP2.

of P on the boundary [Jos05, Lemma 3.6]. Recall that the box (i, j) in the grid label of a

vertex v is shaded if and only if vi ∈ v + Cj . Hence P is the intersection of the halfspaces

v +⋃

i∈A Ci such that v is a vertex of P and A is a minimal subset of columns in the

corresponding grid of v such that the shaded boxes in those column cover all the rows. This

description is redundant in general. We may be able to refine this result as follows.

Conjecture 2.14. In the generic case, a minimal halfspace with respect to P has the form

v +⋃

i∈A Ci where v is a vertex of P, and, in the grid label of v, the shaded boxes in the

columns in A form a partition of [r].

For example, in Figure 2.2 (Example 2.8), no halfspace at v2 is minimal because

no set of columns form a partitions of the rows. However, the sectors 2 and 3 at the point

(0, 3, 2) form a minimal halfspace, and shaded boxes in columns 2 and 3 of the grid form

a partitions of the rows. The converse of the conjecture above is not true, i.e., there are

non-minimal halfspaces of the form described.

Experiments with computation time

We experimented with computing tropical cyclic polytopes Cr,n (which will be

defined in the next section) with r input vertices in n − 1 (projective) dimensions. We

29

used Macaulay 2 [GS02] on a Sun Blade 150 (UltraSPARC-IIe 550MHz) computer with

512MB memory. The computation became infeasible when rn > 80 or so, although r =

30, n = 3 worked. The main problem was the insufficient amount of memory. Some sample

computation times for tropical cyclic polytopes are given in Table 2.2. We see from the

data that computing the Alexander dual can be a problem. This can be made faster using

the software Frobby by Bjarke Roune [Rou07].

n r Initial ideal Alexander dual Free resolution

3 30 74 433 24 21 64 944 236 10 15 221 278 10 70 4169 1106

Table 2.2: Computation times (in seconds) for I, I∗, and the free resolution for tropicalcyclic polytopes Cr,n.

A computation problem we didn’t address is the following:

Question 2.15. How does one obtain a vertex description of a tropical polytope from its

hyperplane description?

2.1.2 Tropical cyclic polytopes

As an application to our method, we will compute and analyze the tropical cyclic

polytopes. Define tropical cyclic polytopes as Cr,n = tconv{v1, . . . ,vr} ⊂ TPn−1, where

vij = (i− 1)(j − 1) for i ∈ [r], j ∈ [n]. Since (i − 1)⊙(j−1) = (i − 1)(j − 1), this is tropical

exponentiation. The Cr,n are generic because the minimum in any k×k minor of the matrix

[vij ] is attained uniquely by the antidiagonal. An example of a tropical cyclic polytope is

shown in Figure 2.5(a).

We will show that the set of 2 × 2 minors of [xij] forms a Grobner basis with

30

respect to V , using the Buchberger Algorithm. If the leading terms of two 2× 2 minors do

not contain a common indeterminate, then both terms in the corresponding S-polynomial

are divisible by one of the original 2×2 minors. Suppose they have the form xijxkl−xilxkj,

and xijxpq − xiqxpj. Then the S-polynomial is xilxkjxpq − xiqxpjxkl. The weights of the

monomials are two of the terms in the 3 × 3 minor of [vij] corresponding to rows i, k, p

and columns j, l, q. The unique minimum of this minor is attained by the antidiagonal.

Therefore the leading term of the S-polynomial cannot be the antidiagonal, so it is divisible

by the leading term of a 2 × 2 minor. Hence the initial ideal I is the diagonal initial ideal

generated by the binomials which are on the diagonals of the 2×2 minors. This correspond

to the staircase triangulation of ∆r−1 ×∆n−1.

(0,0,0,0)

(0,1,2,3)

(0,1,2,3)

(0,2,4,6)

(0,2,4,6)

(0,3,6,9)

��

��

��

��

��

��

��

��

��

��

��

��

��

��

.....

x x

x x x

11 12

x 2r2221

n1 n2 nr

... 1r

...

xxx

... ... .

...

(a) (b)

Figure 2.5: (a) Tropical cyclic polytope on four vertices in TP3. (b) A path correspondingto a generator of the Alexander dual I∗.

Consider a path in an r×n grid representing indeterminates xij , which goes from

the lower left corner to the upper right corner, only moving either right or up at each step

as in Figure 2.5(b). Such paths are precisely the maximal sets, with respect to inclusion,

that do not contain any set {xij , xkl}, i < k, j < l. Hence their complements correspond

31

to the minimal generators of the Alexander dual I∗, which are the monomial labels of the

vertices of Cr,n.

��

��

��

��

��

��

��

��

��

��

��

��

��

��.

��

��

...��

��

..��

(a) (b)

Figure 2.6: (a) Paths in grids corresponding to two one-valent vertices in Cr,n. (b) Hori-zontal and vertical stripes.

The labels of the faces of the tropical cyclic polytope Cr,n are obtained by unshad-

ing the boxes on the paths so that the remaining shaded set still intersects every row and

every column. For example, there are two 1−valent vertices with grids corresponding to

the paths in Figure 2.6(a). The two edges containing these vertices are the only maximal

1−faces, whose labels are obtained by unshading the lower right corner and the upper left

corner, respectively.

We can identify a vertex of Cr,n with the Young diagram above (or below) the

corresponding path in the r × n grid. Then the 1−skeleton of Cr,n is the Hasse diagram of

the Young lattice of the Young diagram fitting in an (r − 1)× (n− 1) grid.

��

��

��

��

��

��

��

��

��

��

��

��

��

��

(a) (b)

Figure 2.7: (a) A diagonal step. (b) Corners indicating that the corresponding monomialsare not minimal.

The shaded part in the label of a k−dimensional face contains the diagonal steps as

32

in Figure 2.7(a) exactly k times because every time we shade in such a corner, the dimension

goes down by one. By straightforward counting, we get that

#k − faces in Cr,n =

(

r + n− k − 2

r − k − 1, n− k − 1, k

)

as seen in Proposition 2.10. That is, out of the r + n− k − 2 steps we take from the lower

left corner to the upper right corner, we take r − k − 1 steps up, n− k − 1 steps right, and

k steps diagonally.

Proposition 2.16. The ordinary generating function for the number Mr,n,k of maximal

k−faces of the tropical cyclic polytope Cr,n equals

∑

r≥1,n≥1,z≥0

Mr,n,kxrynzk =

(

xy

1− y+

yx2

1− x

)/(

1− z

(

xy

1− y−

yx2

1− x

))

.

Proof. A face is maximal if and only if the set of shaded boxes in the r × n grid does not

contain any corners as in Figure 2.7(b). Then Mr,n,k is equal to the number of (k+1)−tuples

of either horizontal or vertical stripes of boxes, as in Figure 2.6(b), such that the sum of

the widths equals n and the sum of heights equals r. The proposition follows from basic

properties of ordinary generating functions.

Moreover, every k−dimensional face contains precisely 2k vertices because every

diagonal step as in Figure 2.7(a) gives 2 ways of shading in the corners, and there are exactly

k such diagonal steps. From this it is easy to see that every k−dimensional face has the

combinatorial structure of a k−dimensional hypercube. Therefore, the f−matrix of Cn,r is

very simple: fk,2k =(

r+n−k−2r−k−1,n−k−1,k

)

, and all other entries are 0.

33

2.2 Cellular resolutions via tropical polytopes

In the previous section, we saw that for a generic tropical polytope, we can con-

struct a monomial ideal whose minimal free resolution is given by the tropical polytope. Not

all monomial ideals arise this way. More specifically, the monomial ideals constructed this

way are precisely the initial monomial ideals of the lattice ideal of a product of simplices.

In this section, we will explore a complementary direction. For an arbitrary monomial

ideal, we will construct a family of cellular resolutions which have a tropical polytope as a

common “shadow”.

Two important operations on monomials are compatible with tropical operations

on the exponent. For monomials xa and xb, their least common multiple is x−(−a⊕−b), and

their product is xa⊙b.

As mentioned in the introduction, tropical discrete geometry arises naturally is as

the image of ordinary discrete geometry over the Puiseux series field with real exponents

K := R((tR)) under the degree map deg : K → R sending an element to its leading (lowest)

exponent. It is naturally an ordered field, where a < b if the leading coefficient of b − a

is positive; its positive elements K>0 comprise the set of all elements with positive leading

coefficient. The field K is real-closed, with t as an infinitesimal. As such, the usual theory

of discrete geometry applies in Kd, and in particular we can define the convex hull of a

point set as the set of all convex combinations of the points. We can also extend the degree

map to deg : Kd → Rd where the degree map is applied coordinatewise. Then the tropical

polytopes are simply the images of objects up above:

Proposition 2.17. Suppose P = tconv(v1, . . . , vk) ⊂ TPd−1, where each vi has first coordi-

34

nate zero. Define lifts vi ∈ deg−1(vi) ∈ Kd>0, and define P = conv({vi}). Then P = deg(P ).

Proof. First, we show that P ⊂ deg(conv({vi})). Suppose that we have a point x =

⊕

ci ⊙ vi ∈ P . Since P ⊂ TPd−1, we can add a constant to each ci such that the largest

ci, without loss of generality c1, is equal to 0. It is easy to lift the ci’s to ci ∈ K with

deg(ci) = ci such that ci > 0 and∑

ci = 1: lift every ci < 0 to tci , and lift ci = 0 to

(1−∑

cj<0 tcj )/|{j | cj = 0}|.

But then we claim that deg(∑

civi) = x. This follows immediately: since there is

no cancellation of the leading terms as all leading coefficients of the ci’s and vi’s are positive,

for each coordinate j we have

deg

(

∑

i

cijvij

)

= max(deg(cijvij)) = max((ci)j + (vi)j) = xj

as desired.

For the reverse direction, using the same logic, it is easy to see that deg(∑

αivi) =

⊕

deg(αi)⊙ vi, again due to the lack of cancellation of leading terms, which shows that the

set deg(conv({v1, . . . , vk})) is a subset of P , completing the proof.

We call the polytope P a lift of P . Note that we can take any lifts of the vertices of

the polytope, as long as the lifted points are in the positive orthant Kd>0; the lift operation

always preserves convex hulls. As an aside, the stipulation that each vi has first coordinate

zero is merely for simplicity; we could ignore this and just look at the facial structure of

the cone generated by the lifts of the vertices. Giving each vi first coordinate zero amounts

to slicing this cone with the hyperplane x1 = 1.

Theorem 2.18. Let P be the tropical polytope given by the convex hull of the points (0,−a),

where a ranges over the exponent vectors of the minimal generators of a monomial ideal

35

I in k[x1, . . . , xd−1]. Take any lift P of P and add the positive orthant in the last d − 1

coordinates, {0}× (K>0)d−1, to P to obtain a polyhedron P

+. Then the complex of bounded

faces of P+

yields a cellular resolution of I.

Proof. We need only to check that for each b = (b1, . . . , bd−1), the bounded faces of P+

with labels dividing b form an acyclic complex. Let the coordinates of the space Kd that

P+

lives in be given by z0, z1, . . . , zd−1, and consider the halfspace H in Kd given by

t−1/2 − tb1z1 − · · · − tbd−1zd−1 ≥ 0. Then a given vertex v ∈ P+

with deg(v) = (0,−a) is in

the halfspace H if and only if xa divides xb. Thus the bounded faces in the halfspace H

are precisely those in the subcomplex X�b. Applying a projective transformation sending

this hyperplane to infinity yields X�b as the complex of bounded faces of some polytope,

which is acyclic. Therefore, each X�b is acyclic, and thus by Proposition 1.1 this complex

yields a cellular resolution of I.

These cellular resolutions can be smaller than the hull resolution. For instance, it

was shown in [Dev04] that no face of the hull complex can have more than (d− 1)! vertices;

this is not true for these cellular resolutions. For example, we can have arbitrarily many

generators of an ideal all lying on the boundary of the same tropical halfspace, and we can

find a lift where these all lay on a single facet, as in the example below.

Example 2.19. Consider the ideal

I = 〈x2z3, xyz3, y2z3, x2y3, xy3z, y3z2, x3y2, x3yz, x3z2, x6, y6, z6〉.

The staircase diagram of I, which is drawn using Gfan [Jen], is depicted in Figure 2.8. The

36

tropical convex hull P of the following points

(0,−2, 0,−3), (0,−1,−1,−3), (0, 0,−2,−3),

(0,−2,−3, 0), (0,−1,−3,−1), (0, 0,−3,−2),

(0,−3,−2, 0), (0,−3,−1,−1), (0,−3, 0,−2),

(0,−6, 0, 0), (0, 0,−6, 0), (0, 0, 0,−6)

corresponding to the twelve generators of I is the negative of the complement of the staircase

diagram inside the 6×6×6 box in R3≥0 including the origin. According to [Dev04], no face of

the hull complex can have more than six vertices. However, notice that the first nine vertices

of P lie on the boundary of the same valid tropical halfspace H with apex (0,−3,−3,−3),

i.e. the first nine vertices lie in the tropical hyperplane defined by the tropical linear form

x0 ⊕ (3⊙ x1)⊕ (3⊙ x2)⊕ (3⊙ x3), and they are all on the boundary between sector 0 and

sectors 1, 2, 3. We can lift the halfspace H to a halfspace H in K4>0, say,

H = {(X0,X1,X2,X3) : −(1 + t + t3)X0 + t3X1 + t3X2 + t3X3 ≥ 0}.

Any lift of the last three points to K4>0 lie in the interior of H, and the following lifts of the

first nine points lie on the boundary of H ∩K4>0:

(1, t−2, 1, t−3), (1, t−1, t−1, t−3 + t−2 − 2t−1 + 1), (1, 1, t−2, t−3),

(1, t−2, t−3, 1), (1, t−1, t−3 + t−2 − 2t−1 + 1, t−1), (1, 1, t−3, t−2),

(1, t−3, t−2, 1), (1, t−3 + t−2 − 2t−1 + 1, t−1, t−1), (1, t−3, 0, t−2),

so these nine points lie on a facet of the lift P+, and the cellular resolution given by this

lift is smaller than the hull resolution.

These lifts include the hull complex as a special case, where each point a is simply

lifted to ta. Like the hull complex, each of them includes the Scarf complex as a subcomplex;

37

Figure 2.8: The staircase diagram of the ideal in Example 2.19

this complex is a simplicial complex where a set of vertices forms a face if the corresponding

set of generators has a unique least common multiple among all LCM’s of sets of generators.

Proposition 2.20. Suppose that a set S of the generators of I has a unique least common

multiple. Then S forms a (simplicial) bounded face in every P+.

Proof. The proof is by induction. We claim that for every v ∈ S, S \ {v} has a unique least

common multiple. Suppose it did not; let T be another set with the same least common

multiple. Then T ∪ {v} has the same least common multiple as S \ {v} ∪ v, so since S has

a unique least common multiple, we must have T ∪ {v} = S. The only other set for which

this is true is T = S, but this contradicts the statement that S has a unique LCM.

Therefore, by induction on the dimension, every proper subset of S forms a face.

Let the least common multiple of S be xb, and as before consider the linear functional

given by f(x) =∑

tbixi. No generator not in S divides xb (otherwise adding it to S gives

the same LCM), and so the hyperplane f(x) = t−1/2 separates the vertices in S from the

vertices outside of S. Again applying a projective transformation to map this hyperplane to

infinity, the induced subcomplex of bounded faces on S must be acyclic. Since it contains

38

every proper subset of S as a face, it must also contain S itself. So S is a bounded face of

P+

as desired.

The following result relates the genericity of tropical polytope to generic mono-

mial ideals defined in [MS05, Chapter 6]. It follows from [MS05, Proposition 6.26] and

Propositions 2.26 and 2.20.

Proposition 2.21. If the negatives of exponent vectors of the minimal generators of a

monomial ideal I are in general position tropically, then I is a generic monomial ideal.

In this case, all the lifts have the same combinatorial type, and the cellular reso-

lution is given by the Scarf complex.

Bayer and Sturmfels referred to the hull complex as a canonical free resolution of

a monomial ideal. Here, however, we see that the hull complex is just one of a family of

resolutions arising from different lifts of the corresponding tropical polytope. For instance, in

Example 2.31 (Figure 2.10), we come up with different free resolutions of the corresponding

monomial ideal. Note that we can always obtain a simplicial resolution by taking a generic

lift. It would be interesting to answer the following questions:

Question 2.22. What does the family of lifts of a given tropical polytope look like? Are

there generalized hull resolutions we can only get by lifting to nonconvergent power series? Is

there a reasonable algorithm for picking a small resolution from among these hull complexes?

Example 2.23. Consider the ideal I = 〈x2z, x2y, xy2z5, xy3z4, xy4z3, xy5z2〉 be an ideal

in the polynomial ring R = k[x, y, z] over any field k. The tropical polytope of interest in

this case is the tropical convex hull of {A,B,C,D,E, F} = {(0,−2, 0,−1), (0,−2,−1, 0),

39

A

A A

B

B

B

C

D

D

E

D

E

F

E

F

Figure 2.9: The three-dimensional part of this is the three cubes in the lower-left corner ofthe figure; it also contains a pair of vertical two-dimensional flaps near vertices C and F,and an extensive top flap connecting vertices A and B to the other vertices.

(0,−1,−2,−5), (0,−1,−3,−4), (0,−1,−4,−3), (0,−1,−5,−2)}. It is depicted in Figure

2.9. Figure 2.10 shows several lifts of this polytope, with different combinatorial types (face

lattices). The shaded faces are those that stay bounded after adding the positive orthant

to the lift P . The first two lifts give free resolutions of the form

0←− I ←− R6 ←− R8 ←− R3 ←− 0

and they are both minimal, although they have different cell structures. The last lift gives

a resolution of the form

0←− I ←− R6 ←− R9 ←− R4 ←− 0

which is not minimal.

A natural question then is:

40

C

E

F

A B

C

D E

F

A

B

C

D E

F

A

B

D

Figure 2.10: Various lifts of the tropical polytope in Figure 2.9. The faces in the back areunshaded.

Question 2.24. Do all cellular resolutions of a monomial ideal occur in this way?

It was shown by Velasco that there exist monomial ideals whose minimal free resolution is

not cellular [Vel]. An affirmative answer to this question may give us a new approach to

this result.

2.2.1 Faces of tropical polytopes

We will now discuss the notion of faces of tropical polytopes. Ultimately we hope

to be able to read off from these the family of cellular resolutions defined above. Since the

facial structure of tropical polytopes is not the main focus of this chapter, the reader is

referred to [DY07] for more discussion.

The idea is that the faces of tropical polytopes should come from the faces of the

lifts to the Puiseux series field. In a sense, each one of these lifts yields a candidate for

the face lattice of the tropical polytope P . The problem is that when the points are not in

(tropically) general position, the lifts can have different combinatorial structures as seen in

Example 2.23.

Let A = [aij] be a d×d matrix whose columns are considered as d points in TPd−1.

The tropical determinant of A is defined by the formula ⊕σ∈Sd(⊙ai,σi

) where Sd denotes

41

the group of permutations of d elements. We say that A is tropically non-singular if the

maximum in its tropical determinant is attained uniquely. In this case, the tropical sign of

A is defined (as in [Jos05]) to be the sign of the permutation that attains the maximum.

Otherwise, the tropical sign is defined to be 0. Let A be a d × d matrix with entries in

K>0 whose degree is A. If the tropical sign of A is not zero, then the sign of the unique

permutation that attains the maximum in the tropical determinant is also the sign of the

leading term of the determinant of A. This observation leads to the following.

Lemma 2.25. For a tropical polytope P with at least d vertices in TPd−1, the oriented

matroid structure of any lift P must refine the partial oriented matroid structure of P given

by the tropical signs on each subset of the d vertices.

On the other hand, there may be a point configuration whose oriented matroid

refines the partial oriented matriod of vertices of P but cannot be obtained as a lift. The

oriented matroid of the tropical polytope in Example 2.23 is the same as that of a square

pyramid with two points at the cone point, but this point configuration cannot be attained

as a lift since distinct points must be lifted to distinct points.

Proposition 2.26. If a tropical polytope P with at least d vertices in TPd−1 is in general

position, then all lifts P are simplicial and have the same oriented matroid structure.

Proof. Let V be a matrix whose columns are vertices of P . The assumption that P is in

general position (in the sense of [BY06, Proposition 4]) implies that all maximal d × d-

submatrices of V are tropically non-singular. By the previous lemma, the tropical signs of

these submatrices determine the chirotope of all the lifts P . Moreover, since these signs are

all non-zero, the lifts are simplicial.

42

It is still possible for lifts of a non-generic tropical polytope to be simplicial and

have the same face lattice. See the tropical (2,4)-hypersimplex in Example 2.35.

For a generic (or simplicial) tropical polytope, every such lift to the Puiseux series

is simplicial and has the same oriented matroid as in Proposition 2.26, hence the same

face lattice. In this case the faces of tropical polytopes should be images of faces of the

lifts. In fact, Joswig’s definition of faces in [Jos05] coincides with these images for generic

tropical polytopes. Unfortunately, when the points are not in generic position, Joswig’s

definition of face lattice has undesirable properties. We propose a different generalization

of this definition to non-generic tropical polytopes as follows:

Definition 2.27. A k-face of a tropical polytope is a minimal k-dimensional subset which

is the union of images of k-faces of every lift under the degree map v.

In other words, the face poset of a tropical polytope is the “common coarsening” of face

posets of all the lifts. The images of faces of lifts under the degree map were called fatoms

(face atoms). The following proposition provides the link between lifted hyperplanes and

tropical hyperplanes.

Proposition 2.28. Let H = {X ∈ Kd : f1X1 + · · ·+fdXd = 0} ⊂ Kd be a hyperplane, and

define H+ := {X ∈ Kd : f(X) ≥ 0}. The image of H under the degree map is a tropical

hyperplane with apex (−deg(f1), . . . ,−deg(fd)), the image of H+ ∩ (K>0)d consists of the

union of the closed sectors indexed by {i : fi > 0}, and the image of H ∩ (K>0)d is the

boundary of this tropical halfspace. (If some fi is equal to 0, then the apex of this tropical

hyperplane has i-th coordinate equal to infinity.)

Proof. Let z = deg(X). If f(X) = 0, then the leading term of f(X) has to cancel. The

43

leading exponent of fiXi is deg(fi) + zi, so the maximum of these d expressions has to

be achieved at least twice. This means that z is in the indicated tropical hyperplane.

Conversely, if z is in the tropical hyperplane, then f(tz) has ties in the leading terms, and

it is trivial to adjust leading coefficients and fill in subleading terms to find a lift of z which

lies on H.

Suppose now that X ∈ H+ ∩ (K>0)d. If z is not in any closed sector indexed by

fi > 0, then each fi for which the maximum of all deg(fi) + zi is achieved has negative

leading coefficient. Since these are the terms which contribute to the leading term of f(X),

f(X) must be negative, a contradiction. Conversely, if z is in a closed sector indexed by,

without loss of generality, f1 > 0, then deg(fi) + zi is maximized by i = 1. Therefore

f1X1 contributes to the leading term of f(X). Finding a lift with sufficiently large leading

coefficient of X1 then ensures that f(X) > 0. This proves the second assertion.

The third statement can be proven the same way as the first, and by noting that

the leading term of f(X) can cancel for X ∈ (K>0)d if and only if deg(X) lies in both the

sectors {i : fi > 0} and sectors {j : fj < 0}.

Armed with this, we can prove a crucial step in this discussion, namely that the

boundary of the lifted polytope indeed maps to the boundary of the tropical polytope.

Proposition 2.29. Let P be a lift of the tropical polytope P ⊂ TPd−1. Then the image of

the boundary of P under the degree map is precisely the boundary of P .

Proof. Every point in the boundary of P lies in some facet. Since the degree map preserves

convex hulls, it must map to a point in the convex hull of the vertices of this facet. However,

by Proposition 2.28, these vertices all lie in a tropical hyperplane which bounds P (namely

44

the image of the hyperplane defining the facet of the lifted polytope), and, since tropical

hyperplanes are convex, the image of our point must also lie on the boundary of P .

For the converse, suppose we have some point v in the boundary of P . This point

is the image of some point in P , since lifts preserve convex hulls; however, a priori, this

point need not lie on the boundary. Consider a tropical hyperplane H with apex v. Not

every open sector of this hyperplane contains a point in P , since that would imply that v

is in the interior of P . Therefore, we can partition the sectors of this hyperplane into a

pair (A,B) such that the A-sectors contain P , with B nonempty. Suppose without loss of

generality that A = {1, . . . , k} and B = {k + 1, . . . , d}. Define a linear functional f on the

lift space via

f = c1t−v1x1 + · · · + ckt

−vkxk − t−vk+1xk+1 − · · · − tvdxd, (2.1)

where the ci’s are positive real constants. Since P lies in the union of the A-sectors of H,

every vertex w of P has wi − vi maximized for some i ∈ [k]. Therefore, the leading term of

the expression given by f(w) contains some positive summand from among the first k terms

of f . So, if we make the ci’s large enough, we can ensure that f(w) > 0 for all vertices w

of P . Fix a set of such ci’s.

But we can easily lift the point v to a point v with f(v = 0); for instance, simply

taking vi = 1ci

tvi for i ∈ [k] and vi = kd−k tvi otherwise. Hence v has a lift which lies

outside of P . So, we have constructed a lift of v which lies inside of P , and a lift which

lies outside of it. The line segment between these two lifts, each of which has positive

leading coefficients, consists entirely of other lifts of v, and must intersect the boundary of

P somewhere, completing the proof.

45

The degree map is well-behaved; for instance, it also does not increase the dimen-

sion of a face.

Proposition 2.30. If F is a k-face of P , then the image of F under the degree map is at

most k-dimensional.

Proof. Suppose we have a k-face F of P . Triangulate this with its original vertex set,

dividing it into a number of k-simplices which are the convex hulls of k + 1 lifts of vertices

of P . The image of each such simplex under the degree map, by Proposition 2.17, is the

convex hull of k+1 points in TPd−1, and is therefore at most k-dimensional (see e.g. [DS04].)

Therefore, the image of F under the degree map is the finite union of k-dimensional things,

and is therefore k-dimensional.

Example 2.31 (The model). Let P be the tropical convex hull of

(A,B,C,D,E, F ) = (0201, 0210, 0125, 0134, 0143, 0152) ⊂ TP3,

as seen in Example 2.23.

This is a tropical 3-polytope lying in three-space, shown in Figure 2.9. Its nat-

ural polyhedral decomposition dual to a subdivision of ∆5 × ∆3 consists of three three-

dimensional cells, all unit cubes, and a number of two-dimensional flaps. It has 29 vertices,

clearly too many for the tropical convex hull of 6 points.

The facets defined in [Jos05], which we will call J-facets, are ABDC, ABED, ABFE,

ABFC, and CDEF. They exhibits a number of the problems. For example, the facet ABED

is not an extreme set, i.e. we can find two points not on the facet but whose convex hull

intersects the facet (Figure 2.11). The fundamental problem here is that in some lifts,

46

��

��

A

B

C

E

F

D

x

y

z

Figure 2.11: The J-facet ABED, and two points which demonstrate that it is not an extremeset.

ABED itself is not a facet of the lift. In those lifts, we can take two points not in the lift

of ABED for which the tropical line segment connecting them pierces the convex hull of

ABED.

Moreover, the facets ABDC, ABED, ABFE, and ABFC intersect badly. To be

precise, their intersections are two-dimensional; the convex hulls of ABDC and ABED

intersect in the convex hull of ABD, a two-dimensional object. Furthermore, according to

Joswig’s definition, A and B are not actually vertices of this polytope. If we intersect the

facet vertex sets setwise, we never get the singletons A and B, but merely the atom AB. So

according to Joswig’s definition, this object is a pyramid over a square, with vertices AB

(apex), C, D, E, and F. But this misses some aspects of the tropical polytope, as A and B

are both vertices in the sense that if we remove either, the convex hull changes. There are

also examples where the J-face lattice does not have correct height, as in Example 2.32.

By our new Definition 2.27, the tropical polytope P has three facets; the upper

47

C F

A B

D E

A B C D E F

CDEFABCFunderbelly

edges (L to R): AB, BF, AC, CD, DE, EF, CF

Figure 2.12: The face lattice of the model (L) and an artist’s rendition of it as a cell complex(R). The latter consists of a flat hexagon with a couple of squares puckering up.

shell comprising the convex hull of {A,B,C, F}, the lower face comprising the convex hull of

{C,D,E,F}, and the “underbelly” given by the union of the convex hulls of {A,B,C,D},

{A,B,D,E}, and {A,B,E, F}. The problematic pairs of faces which intersect improperly,

such as (ABCD,ABDE), have all been combined into the underbelly. This underbelly is a

facet because in every lift, it is the union of facets of the lifted polytope, and it is minimal

with this property. The face poset consists of three 2-faces, seven 1-faces (AB, AC, CD,

DE, EF, CF, BF), and six 0-faces (the vertices); see Figure 2.12.

Example 2.32. Let P be the tropical convex hull of

(A,B,C,D,E, F,G,H, I) = (0301, 0310, 0224, 0233, 0242, 0158, 0167, 0176, 0185) ⊂ TP3.

This polytope (Figure 2.13) is merely a three-tiered version of the two-tiered pre-

vious example. The J-facets are ABFI, ABDC, ABED, CDEGF, CDEHG, and CDEIH. By

Joswig’s definition, taking setwise intersections of these J-facets, we have a chain of faces

given by D ⊂ CDE ⊂ CDEG ⊂ CDEGF . This chain is too long to live in the boundary

of a 3-polytope; the face poset does not have the correct height.

If P is in general position, by Proposition 2.26, the face lattice of P is the same as

48

Figure 2.13: A tropical 3-polytope which is a more complicated version of the model.

that of any lift. In this case, the face lattice is determined by combinatorially intersecting

the vertex sets of facets, and this new definition is the same as that of Joswig’s. In essence,

this definition gets rid of many of the problems with Joswig’s definition in the non-generic

case via the following method: whenever two J-facets fundamentally intersect improperly,

they are declared to be part of the same facet of P . Note also that it is clear that the 0-faces

of P are always precisely the vertices.

These and other examples lead us to formulate the following conjectures about our

new definition, each of which represents an improvement on Joswig’s.

Conjecture 2.33.

1. The k-faces of tropical polytopes are extreme sets, i.e. there should not be two points in

the polytope not in a face such that the tropical line segment connecting them intersects

the face.

2. The (topological) boundary of a k-face is a union of (k − 1)-faces. Thus, the faces fit

49

together to form a cell complex.

3. The homology of this cell complex is that of a sphere.

4. The intersection of two faces is well-defined (i.e. does not depend on the lift), and is

itself a contractible union of faces. Note that in some cases, a (for instance) 3-face

in a lift which maps to a two-dimensional object under the degree map may be part of

the lifts of two 3-faces of P . In this case, though, its contribution to the intersection

will still be two-dimensional.

5. k-faces of tropical polytopes are always contractible.

6. The faces of a tropical polytope do not depend on the provided vertex set. In other

words, if we consider P as the convex hull of a different set of points (and form lifts

by lifting this different set of points), the k-faces of P will be the same for all k.

We now present two more examples of tropical polytopes.

Example 2.34. Let P be the convex hull of {A,B,C,D,E} = (0101, 0011, 0002, 0001, 0110) ⊂

TP3, as depicted in Figure 2.14.

Discussion: This is a cube with a pendant edge. We previously encountered the

convex hull of the vertices ABC; this tropical polytope consists of the union of three facets

of a cube surrounding a vertex (0112 here). Adding the point D realizes the unit cube as

a tropical polytope; its facets are simply the convex hull of ABC, which consists of three

facets of the cube, and the convex hulls of ABD, ACD, and BCD, each of which is another

facet of the cube. This is a perfectly normal tropical tetrahedron (albeit not one in general

position.) All lifts of this polytope are simplices.

50

x

y

z

B

C

D

E

A

Figure 2.14: A five-vertex tropical polytope in TP3: cube with pendant edge.

C

D

A

BE

C

D

A

BE

C

D

EA

B

C

E

D

A

B

Figure 2.15: Four lifts of the cube with pendant vertex. Two are bipyramids and two aresquare pyramids.

51

DE

A

B

C

A B C D E

ACD BCD AEBC AEBD

edges (L to R): AC, BC, AD, BD, CD, AE, BE

Figure 2.16: The face lattice of a cube with pendant edge (L) and its realization as a cellcomplex (R), a tetrahedron with one edge subdivided.

Adding the pendant vertex, however, produces a variety of possible lifts. There

exist lifts where the lifted point E lies in various places with respect to the tetrahedron

ABCD; for instance, it could be coplanar with 0, e1, and e2 (as these four points lie on

a tropical hyperplane), or with e1, e2, and e3. Even if it is in general position, it can be

beyond different faces of the tetrahedron. See Figure 2.15 for some possibilities.

In all, the face lattice of this tropical 3-polytope is depicted in Figure 2.16. This

is a tetrahedron formed by ABCD, with the edge AB subdivided by point E. It has four

facets: two triangles ACD and BCD, and two squares ABCE and ABDE. The two squares

intersect in the union of the edges AE and BE. Its f-vector is (5, 7, 4). The fact that point

E subdivides edge AB is interesting; intuitively, this makes sense, since the point through

which point E is connected to the rest of the polytope lies on edge AB. If we had placed

point E on point AB, the diagram would be the same, except that point E would not appear

as a vertex at all.

Example 2.35. Let P be the convex hull of {A,B,C,D,E, F} = (0011, 0101, 0110, 1001, 1010, 1100) ⊂

52

x

y

z

Figure 2.17: A tropical (2, 4)-hypersimplex, i.e. octahedron.

TP3, as depicted in Figure 2.17.

Discussion: This polytope is an instance of a tropical hypersimplex, to be precise

the (2, 4)-hypersimplex, whose vertex set consists of all 4-tuples with two 1’s and two 0’s.

These (n, d)-hypersimplices always lie on a hyperplane, to be precise one with apex 0.

However, their hull lifts are bona fide hypersimplices, and in particular are full-dimensional;

their face lattice will accordingly be d-dimensional. Indeed, it turns out that every lift of

this object is a (2, 4)-hypersimplex, which is an octahedron, so its face lattice is simply that

of an octahedron. In general, the face lattice of the tropical hypersimplex is identical to

that of the corresponding ordinary hypersimplex.

Interestingly, four of the facet-defining tropical hyperplanes are the same (the one

with vertex 0); this hyperplane cuts out different facets by virtue of taking different sectors

to comprise the relevant halfspace.

53

Chapter 3

Tropical Linear Spaces

Tropical linear spaces are tropical analogues of usual linear spaces. The tropical

hyperplane of a tropical linear form (c1 ⊙ x1) ⊕ · · · ⊕ (cn ⊙ xn), c ∈ TPn−1, is the set of

points x ∈ TPn−1 such that the minimum in the linear form is attained at least twice. Let

us recall the setup from [Spe]. Let d ≤ n be positive integers. A point p ∈ TP([n]d )−1 is

called a tropical Plucker vector if for every d− 2-subset S of [n] and four distinct elements

i, j, k, l ∈ [n]\S, the minimum in (pSij ⊙ pSkl)⊕ (pSik ⊙ pSjl)⊕ (pSil ⊙ pSjk) is attained at

least twice. Given a tropical Plucker vector p ∈ TP([n]d )−1, for each d + 1-subset I ⊂ [n] we

can define a tropical linear form ⊕i∈I

(

pI\{i} ⊙ xi

)

called a circuit. The tropical linear space

corresponding to p is the intersection of the tropical hyperplanes of these circuits. A tropical

basis of a tropical linear space is a set of defining linear forms for the space. Tropical bases

are not unique, and need not be minimal in any sense. Much of this chapter is concerned

with finding minimal tropical bases.

A tropical linear space whose defining tropical linear forms have coefficients all 0

54

or ∞ is called constant coefficient. Its associated tropical hyperplanes are determined by

the supports of the tropical linear forms, that is, the entries with non-∞ coefficients. As

a result, the conditions for being a tropical basis depend only on those supports. In this

case, we can deal with the matroid whose dependent sets are supports of these tropical

linear forms. In Sections 3.1 and 3.2, we deal exclusively with the constant coefficient case,

which amounts to finding tropical bases of matroids. We describe minimal tropical bases for

several classes of matroids. Our main findings, appearing in Section 3.2, are that graphic

matroids, cographic matorids, and the matroid R10 have unique minimal tropical bases. We

hope to extend these findings to all regular matroids.

In Section 3.3, our main result concerns the tropical rank of a matrix whose rows

form a tropical basis. Furthermore, we conjecture a criterion for being a tropical basis in the

non-constant case. Finally, in Section 3.4, we show that there is a natural parametrization

of tropical linear spaces in terms of cocircuits. This chapter is based on joint work with

Debbie Yuster [YY].

3.1 The constant coefficient case

As discussed in the Introduction, constant coefficient tropical linear spaces can

be described in terms of their associated matroids. We will deal with the matroid whose

circuits are supports of circuits of a tropical linear space.

Let M be a matroid on the ground set {1, 2, . . . , n} and C be the collection of

its circuits. For a circuit C ∈ C, let T (C) be the set of points x ∈ TPn−1 such that the

minimum value in {xi : i ∈ C} is attained at least twice. The set T (C) :=⋂

C∈C T (C) is a

55

polyhedral fan called the tropical variety or the Bergman fan of M . Given a subset B ⊂ C,

define T (B) :=⋂

C∈B T (C). The set B is called a tropical basis of M if T (B) = T (C). We

wish to study the following problem.

Problem 3.1. Identify a minimal tropical basis for any matroid.

It was shown in [AK06] that the intersection of the tropical variety of a matroid

M and a sphere centered at the origin is a geometric realization of the order complex of

the lattice of flats of M . In a matroid, an element e is called a loop if {e} is a circuit, and

two elements e1, e2 are said to be parallel if {e1, e2} is a circuit. Since removing the loops

and replacing each parallel class with a single element in a matroid does not change the

lattice of flats, we may assume that our matroids are simple, i.e. contain no loops or parallel

elements. Since the circuits of the direct sum (or 1-sum) of two matroids is the union of

circuits of the summands, the following is clear.

Lemma 3.2. If matroids M , M1, and M2 are such that M = M1 ⊕M2, then the tropical

bases of M are precisely unions of tropical bases of M1 and M2.

Since every matroid is the direct sum of its connected components, we can restrict

attention to connected matroids.

Each circuit in the tropical basis “excludes” certain points from being in the trop-

ical variety, namely those values which induce a unique minimum on the terms of that

circuit. We can cut down a tropical basis of a matroid as long as the smaller circuit set

excludes the same points as the larger one. In comparing the excluded points, it suffices to

consider the 0/1 points:

56

Lemma 3.3. For any B ⊂ C,

[T (B) ∩ {0, 1}n = T (C) ∩ {0, 1}n] =⇒ [T (B) = T (C)].

In other words, to test equality of T (B) and T (C), it is sufficient to check that they agree

on 0/1 points.

Proof. Suppose we are given a subset B ⊂ C of circuits of a matroid M , and we would like

to know if B is a tropical basis of M . Since each circuit of a matroid excludes points, T (B)

contains, or is equal to, T (C). Thus, we need only worry about “extra” points, i.e. points

which are in T (B) but not in T (C). We show that if T (B) contains an extra point x, then

T (B) contains an extra 0/1 point. Thus checking that there are no extra 0/1 points shows

equality of T (B) and T (C).

Suppose T (B) contains an extra point x. Then there exists a circuit C ∈ C\B

which excludes x, and therefore the set {xi : i ∈ C} has a unique minimum m. From x,

construct the 0/1 vector v as follows:

vi =

1 if xi > m

0 if xi ≤ m

We see that v is excluded by C, so v /∈ T (C). However, for any circuit C such that

{xi : i ∈ C} attains its minimum at least twice, {vi : i ∈ C} also attains its minimum at

least twice, so v ∈ T (B).

So it is possible to remove some circuits from a tropical basis of a matroid, if the

same 0/1 vectors are excluded in the smaller set. It was shown in [AK06] that a 0/1-vector

is in the tropical variety if and only if its support, the set of coordinates with non-zero

57

values, is a flat of the matroid. Hence the excluded points are non-flats. We conclude that

a collection of circuits is a tropical basis if and only if it excludes all 0/1 non-flats.

If two circuits C1, C2 have a unique element in their intersection, pasting them

means taking their symmetric difference C1△C2 = (C1\C2) ∪ (C2\C1).

Lemma 3.4. If a collection S of circuits of a matroid has the property that every other

circuit of the matroid can be obtained by successively pasting circuits in S, then S is a

tropical basis.

Proof. Suppose we have a weight assignment x ∈ Rn such that all circuits in S have their

minimum value attained at least twice. Consider the circuit C obtained from pasting A

and B together along some element e, where A,B ∈ S. We claim that C too attains its

minimum at least twice.

The minima may be attained in one of three ways:

Case 1) A attains its minimum on a and a′, neither of which is e, and B attains its

minimum on b and b′, neither of which is e. In this case, C attains its minimum twice, on

either a and a′, or b and b′ (or all four), depending on which element weights are minimal.

Case 2) A attains its minimum on elements a and e and B attains its minimum on

edges b and b′, neither of which is e. Then b and b′ have lower weight than e, so C attains

its minimum on b and b′.

Case 3) A attains its minimum on elements a and e, and B attains its minimum on

edges b and e. Then weight(a) = weight(e) = weight(b), and thus C attains its minimum

on a and b.

58

3.1.1 Partition matroids

Given a partition Π of [n], the partition matroid MΠ is the matroid whose circuits

are pairs of elements in the same block of the partition [Oxl92]. This is the case when the

defining ideal of the linear space is also binomial.

A minimal tropical basis consists of enough pairs in each block to form a “spanning

tree” on that block. This ensures uniform weighting within each block, and thus forces the

minimum to be attained twice on each circuit.

3.1.2 Uniform matroids

The uniform matroid Ud,n is the matroid arising from a generic set of n points

in Rd. The circuits of Ud,n are the (d + 1)-subsets of [n]. Any tropical basis of Ud,n must

contain at least 1d+1

(

nd

)

elements [BJS+, Theorem 2.10], and the bound is not tight.

Lemma 3.5. An inclusion minimal tropical basis B for a uniform matroid U is given by

B = {C : i ∈ C},

where i is any fixed element of the matroid.

Proof. We must show that given a point x ∈ Rn that is excluded by some circuit of U , we

can find a circuit in B which excludes x. As shown above, we can restrict our attention to

0/1 points. Without loss of generality let us fix i = 1, so that the tropical basis B consists

of all circuits of U containing the element 1. Consider a 0/1 point x ∈ Rn that is excluded

by some circuit Cx. The minimum of {xj : j ∈ Cx} is attained uniquely. Consequently,

there is exactly one element k of Cx such that xk = 0.

59

If Cx contains the element 1, then it is in B and we’re done. Otherwise, consider

the following two cases:

Case 1 (x1 = 1): Let Cx be the circuit obtained from Cx by replacing any element

of Cx other than k by the element 1.

Case 2 (x1 = 0): Let Cx be the circuit obtained from Cx by replacing k with 1.

In both cases, Cx is a circuit of U , since every (d + 1)-subset of [n] is a circuit.

It contains the element 1, so it is in B. Finally, Cx excludes x, since the minimum of

{xj : j ∈ Cx} is attained uniquely. Thus x is excluded by the tropical basis B.

To see that B is inclusion minimal, consider a circuit C∗ ∈ B. The point x, where

xi = 1 for all i ∈ C∗\1 and xi = 0 otherwise, would be excluded by B, but not by B\{C∗}.

Thus it is necessary to include C∗ in this tropical basis.

Note that the tropical bases given for uniform matroids are not unique. For ex-

ample, consider U2,4. It has four circuits, namely 123, 124, 134, and 234. Any collection

of three out of the four circuits forms an inclusion- and cardinality- minimal tropical basis.

Yet, there is no single circuit that must be in a tropical basis of U2,4. Additionally, the

tropical bases given above for uniform matroids are not in general cardinality-minimal, de-

spite being inclusion-minimal. For example, the smallest tropical bases of U2,5 contain five

elements (123, 124, 125, 134, and 345 for example), however there are inclusion-minimal

tropical bases containing six elements, namely the six circuits which contain the element ‘1’.

60

1

2 3

4 7

5

6

Figure 3.1: Fano plane

3.1.3 Fano plane

The Fano plane gives rise to a matroid whose ground set consists of its 7 vertices.

The circuits are the “lines” in the point configuration (see Figure 3.1) and any 4-subset not

containing a line.

The unique minimal tropical basis for the Fano plane consists of the seven 3-

element circuits, one arising from each line (124, 137, 156, 235, 267, 346, & 457).

The lines are necessary: If one of the lines, L, were to be excluded, we could assign

weight 1 to two of the line’s points, weight 0 to the remaining point, and weight 0 to all

points off the line. Thus every circuit except L would have its minimum attained twice,

while L would have a unique minimum.

The lines are sufficient: Each 4-element circuit (1236, 1467, 2456, 3567, 1257,

1345, and 2347) can be obtained by pasting two 3-element circuits together along a shared

element, and deleting that element. For example, we can think of the circuit 1236 as being

pasted from the circuits 124 and 346. By Lemma 3.4, the 3-element circuits are sufficient.

Problem 3.6. Find a minimal tropical basis for transversal matroids [STY06, Section 3.3].

61

Problem 3.7. What happens to tropical bases under deletion and contraction?

3.2 Regular matroids

A regular matroid is one that is representable over every field. In this section we

characterize minimal tropical bases of certain important classes of regular matroids, namely

graphic matroids, cographic matroids, and the matroid R10. Seymour showed that every

regular matroid can be constructed by piecing together matroids of these types [Sey80].

3.2.1 Graphic matroids

A graphic matroid is formed from a graph G. The edges of G form the ground set,

and the circuits of the matroid are the minimal edge collections corresponding to cycles of

G, where a cycle is a closed walk all of whose vertices have degree two in the cycle. If G

contains n edges, we can think of a point in Rn as giving edge weight assignments for the

edges of G. A point in Rn is in the tropical variety of the graphic matroid arising from G

if and only if each cycle in G attains its minimum edge weight at least twice. We restrict

our attention to graphic matroids arising from graphs with no loops and no parallel edges.

Theorem 3.8. The unique minimal tropical basis for a graphic matroid is the collection of

its induced cycles.

An induced cycle is an induced subgraph that is itself a cycle.

Proof. Induced cycles are sufficient: Suppose we have a weight vector which achieves its

minimum twice on each induced cycle. Consider a circuit arising from a non-induced cycle

C. Then the induced subgraph on the vertices contained in C contains a chord. Divide

62

C into cycles C1 and C2 along this chord. Continuing in this manner, we can decompose

C into a collection of induced cycles pasted together. Thus by Lemma 3.4, induced cycles

form a tropical basis.

Induced cycles are necessary: Suppose we have an induced cycle C which is not in

our tropical basis. Then it is possible to construct an edge weighting for which the minimum

is attained at least twice on every cycle except C. Assign all but one edge of C weight 1.

Assign weight 0 to the remaining edge of C and to all other edges of the graph. Every

cycle incident with C has two or more edges that are not contained in C, since otherwise

C would contain a chord and thus not be an induced cycle. Thus all cycles except C have

their minimum attained at least twice, while C has a unique minimum. Thus it is necessary

to have C in every tropical basis.

3.2.2 Cographic matroids

A cographic matroid is formed from a connected graph G, having the edges of G as

its ground set. Circuits of cographic matroids are the inclusion-minimal edge cuts, i.e., sets

of edges such that removing them makes G disconnected. In order to insure every circuit

contains at least 3 elements, we will restrict our attention to cographic matroids arising

from 3-edge-connected graphs. A graph is called k-edge-connected if it remains connected

after removing any k − 1 edges. A bridge of a connected graph is an edge whose removal

disconnects the graph. A connected graph is 2-edge-connected if and only if it does not

contain a bridge.

Theorem 3.9. The unique minimal tropical basis of a cographic matroid M on a graph G

63

V

0

0

1 1 1 0

UA B

Figure 3.2: 2-edge-connected graph

consists of the edge cuts that split G into two 2-edge-connected subgraphs.

Proof. For an edge cut C that splits G into connected subgraphs U and V , let the index of

C be the sum of the number of bridges in U and V . We will show by induction on the index

that the circuits of positive index can be obtained by successively pasting index 0 circuits,

as in Lemma 3.4.

Suppose an edge cut C splits G into connected subgraphs U and V . Let e be a

bridge in U that splits it into subgraphs A and B. Let brA, brB , and brV be the number of

bridges in A, B, and V respectively. Then the index of C is equal to 1 + brA + brB + brV .

Let C ′ be the edge cut that splits A and B ∪ V . Here B ∪ V denotes the induced subgraph

of G on the vertices in B and V . In B ∪ V , there are at least two edges between B and V

because otherwise the edge e and the unique edge would disconnect G, contradicting the

3-edge-connectedness of G. Hence a bridge in B ∪ V must be either a bridge in B or a

bridge in V . Therefore, the index of the edge cut C ′ is at most brA + brB + brV . Similarly,

the index of the edge cut C ′′ that splits B and A∪ V is at most brA + brB + brV . The edge

64

cut C is obtained by pasting the lower index circuits C ′ and C ′′ along e. By induction on

the index, we see that C is obtained by successively pasting index 0 circuits. This proves

(by Lemma 3.4) that the index 0 circuits form a tropical basis.

Now we show that all such edge cuts are necessary. Suppose an edge cut C ⊂

edges(G) splits the graph G into subgraphs U and V , each of them 2-edge-connected.

Consider edge weights as follows: each edge contained in U and V gets weight 0, one edge

going between U and V gets weight 0, and the other edges between U and V get weight 1.

Consider any other edge cut C ′ 6= C. It must cause either U or V to become disconnected.

Suppose it cuts U into subgraphs A and B (see Figure 3.2). Since U is 2-edge-connected,

there are at least 2 edges going between A and B, and those edges are in C ′. Hence the

minimal weight 0 is attained at least twice on C ′, but not on C. Thus C must be in our

tropical basis.

3.2.3 The matroid R10

The matroid R10 is a regular matroid whose elements are given by the edges of

the complete graph on five vertices, K5. Its circuits are given by the fifteen 4-cycles of K5,

along with the complement of each 4-cycle. Thus R10 has 30 circuits.

Proposition 3.10. The unique minimal tropical basis of R10 consists of the fifteen 4-cycles.

Proof. The 4-cycles are necessary: For any 4-cycle C = {a1, a2, a3, a4}, we can construct a

0/1-point x which is excluded by C, but not any other circuit of R10. Set xa1 = xa2 = xa3 =

1, and assign 0 to all other entries of x. Then x attains its minimum value on C uniquely,

while on every other circuit its minimum is attained at least twice. This is because no other

65

4-cycle can share 3 elements with C, and any 4-cycle-complement will contain at least three

elements disjoint from C, corresponding to ‘zero’ entries of x.

In order to show the 4-cycles are sufficient, we show that any 0/1-point excluded

by a 4-cycle-complement will also be excluded by a 4-cycle. Consider a point x which is

excluded by a 4-cycle-complement C = {a1, a2, a3, a4, a5, a6}. Since x attains its minimum

uniquely on the terms of C, without loss of generality suppose xa1 = 0 and xa2 = xa3 =

xa4 = xa5 = xa6 = 1. Note that no 4-cycle-complement contains a 4-cycle. In order for a

4-cycle to exclude x, it must intersect C in exactly three elements other than a1. All 4-cycle-

complements look the same up to permuting the vertices. Figure 3.3 shows one such picture.

One can easily see that for any 5-element subset of the 6 edges, there exists a 4-cycle meeting

the subset in exactly 3 edges. Therefore every point excluded by a 4-cycle-complement is

also excluded by a 4-cycle.

Figure 3.3: 4-cycle-complement

Problem 3.11. Do all simple regular matroids have unique minimal tropical bases?

The Fano plane is not regular but has a unique minimal tropical basis.

66

3.3 Non-constant coefficient case

We now discuss the non-constant coefficient case. A matroid polytope of a rank

d matroid on the ground set [n] is the convex hull of the characteristic vectors in Rn of

the bases of the matroid. The hypersimplex ∆d,n is the convex hull of 0/1 vectors in Rn

with coordinate sum d. A matroidal subdivision of ∆d,n is a regular subdivision in which

every face is a matroid polytope. It was shown in [Spe] that a point p ∈ TP([n]d )−1 is a

tropical Plucker vector if and only if it induces a matroidal subdivision of ∆d,n. Moreover,

the corresponding tropical linear space is the polyhedral subcomplex of the dual of the

subdivision, consisting of the cells dual to matroid polytopes of loop-free matroids, i.e.

matroids without any one-element circuits.

A coordinate of p being ∞ is equivalent to deleting the corresponding vertex from

the hypersimplex ∆d,n. The case when all coordinates of p are either 0 or ∞ is the constant

coefficient case discussed in Sections 2 and 3. In this case, the subdivision contains a single

matroid polytope, and the circuits of the corresponding matroid are precisely the supports

of the circuits defined in the Introduction. The tropical linear space of a matroid is also

called the Bergman fan.

Since the hypersimplex ∆d,n lies on the hyperplane of points whose coordinates

sum to d, the duals of the subdivisions, hence the tropical linear spaces, have a lineality

space containing R(1, . . . , 1), as seen above. Since each cell in a matroidal subdivision is a

matroid polytope, the star of any face in the tropical linear space is isomorphic to a constant

coefficient tropical linear space.

Let K be the field of Puiseux series∑

a∈I caxa where I is a locally finite subset of

67

R with a least element and ca ∈ C. Let the degree map deg : K → T = R∪{∞} be the map

that sends an element in K\{0} to its leading (lowest) exponent of x and sends 0 to ∞. It

induces a map from a projective space over K to a tropical projective space. Let L be a d-

dimensional vector subspace in Kn, and let P ∈ P([n]

d )−1

K be its Plucker vector, i.e. the vector

of maximal minors of a matrix whose row space is L. Then deg(P ) is a tropical Plucker

vector. If a tropical Plucker vector arises in this way, then the corresponding tropical

linear space coincides with the image deg(L) ⊂ Tn and is called realizable. If there is a

representative Plucker vector containing only complex numbers, then we get the realizable

constant coefficient case. For a linear form f = a1X1 + · · · + anXn with ai ∈ K, let its

tropicalization be the tropical linear form (deg(a1)⊙x1)⊕· · ·⊕ (deg(an)⊙xn). The circuits

defined in the Introduction are precisely the tropicalizations of linear forms with minimal

support in the orthogonal complement of L. We will refer to a set of linear forms over Kn

as a tropical basis if their tropicalizations define the tropical linear space.

The tropical determinant of an r × r square matrix A = [aij ] is defined to be

⊕

σ∈Sr

(a1σ1 ⊙ · · · ⊙ arσr), where Sr is the symmetric group of order r!.

The tropical rank of a matrix is the largest integer r such that there exists a tropically non-

singular submatrix of size r, i.e. the minimum in the tropical determinant is not unique.

In the special case of realizable tropical linear spaces all of whose tropical Plucker

coordinates are non-∞, we establish a necessary condition for a set of linear forms to form

a tropical basis, which generalizes Theorem 5.3 in [RGST05].

Theorem 3.12. Let L be a n− k dimensional linear subspace in Kn all of whose Plucker

coordinates are non-zero. Let M ∈ Km×n,m ≥ k, be a matrix whose rows are non-zero

68

elements in the orthogonal complement of L. If the rows of M form a tropical basis for L,

then any k columns of deg(M) have tropical rank k.

Proof. Suppose that there is a m× k submatrix A of deg(M) with tropical rank less than

k. Then by [DSS05, Theorem 5.5] the Kapranov rank of A is less than k, which means that

there is an m×k matrix A′ over K with rank less than k such that A = deg(A′). Let v ∈ Kk

be a non-zero vector in the kernel of A′. Then deg(v) is in the tropical prevariety in Tk

defined by the m rows of A. We can augment deg(v) to a vector in the tropical prevariety

of the rows of deg(M) by putting ∞ in the other n − k coordinates. The support of this

vector has size at most k, however, the points in the tropical linear space have support size

at least k + 1 because of the hypothesis that all the tropical Plucker coordinates are finite.

Hence the prevariety is not equal to the tropical linear space, so the rows of M do not form

a tropical basis.

Conjecture 3.13. The converse of Theorem 3.12 holds.

Example 3.14. The proposition and the conjecture do not apply when some of the tropical

Plucker coordinates are ∞. Consider the 2-dimensional linear subspace of K4 which is the

kernel of the matrix

M =

1 0 1 1

0 1 1 1

1 −1 0 0

, whose degree is

0 ∞ 0 0

∞ 0 0 0

0 0 ∞ ∞

.

The circuits are precisely these three rows, hence the 0/1 points in the tropical variety are

(0, 0, 0, 1), (0, 0, 1, 0), (1, 1, 0, 0). The last two columns of deg(M) have tropical rank only 1,

but the rows of M form a tropical basis. Notice that the corresponding tropical Plucker

69

coordinate is ∞. The first two rows of the matrix deg(M) have the same tropical rank as

the whole matrix in every subset of columns, but they do not form a tropical basis. Hence

we cannot determine whether a set of linear forms is a tropical basis just from the tropical

ranks of columns.

In the cases when Theorem 3.12 and its converse are applicable, we would get an

algorithm for checking if a given matrix is a tropical basis.

3.4 Parametrizations of tropical linear spaces

So far we have been looking at tropical linear spaces as intersections of tropical

hyperplanes. In this section, we will look at them as images of tropical linear maps.

Let A be an n × d matrix over K whose image (column space) is L. Let deg(A)

be the matrix whose entries are the degrees of entries in A. This matrix defines a tropical

linear map deg(A) : Td → Tn, v 7→ deg(A) ⊙ v, where the tropical matrix multiplication ⊙

is defined by replacing sums with minima and products with sums in the evaluation of the

ordinary matrix product.

For any such A, we have

deg(L) = deg(im(A)) ⊇ im(deg(A)) (3.1)

The containment holds because the columns of deg(A) are in the tropical linear space

deg(L), and so are their tropical linear combinations, since tropical linear spaces are closed

under taking tropical linear combinations. Similar expressions hold in much more generality

for tropical varieties, as shown in [PS04, Theorem 2]. We are interested in knowing when

the equality deg(L) = im(deg(A)) is attained.

70

A cocircuit of the linear space L is an element in L whose support is minimal

with respect to inclusion. They are circuits of the orthogonal complement L⊥. Two cocir-

cuits with the same support must be constant multiples of each other, since otherwise one

coordinate can be cancelled to get a vector with a smaller support.

Lemma 3.15. Every nonzero v ∈ L can be written as v = v1 + · · ·+ vd for some cocircuits

vi ∈ L such that deg(v) = deg(v1)⊕ · · · ⊕ deg(vd).

Proof. If v is a cocircuit, then we are done.

Suppose not. Let u ∈ L be a cocircuit with supp(u) ⊂ supp(v). Then for c ∈ K

with large enough degree, we have deg(cu) ≥ deg(v) coordinatewise. Pick such a c so that

cu and v coincide in at least one coordinate, i.e. supp(v−cu) ( supp(v). Let v1 = cu. Since

deg(v1) ≥ deg(v), we have deg(v) = deg(v1)⊕ deg(v − v1).

If v−v1 is not a cocircuit, then we can repeat the same argument on v−v1, which

has a strictly smaller support. We will end up with cocircuits with the desired property.

Theorem 3.16. The equation deg(L) = im(deg(A)) holds if and only if every cocircuit in

L is represented in A.

Proof. The “if” direction follows immediately from previous Lemma. For the “only if”

direction, suppose there is a cocircuit c ∈ L whose support is not represented in A. Then

deg(c) ∈ deg(L)\im(deg(A)) since any vector in im(deg(A)) with finite coordinates in

supp(c) also has finite coordinates outside supp(c).

This theorem for the constant coefficient case appeared in [DSS05, Proposition 7.5].

71

Chapter 4

Implicitization and Elimination

Implicitization and elimination are fundamental operations in computational al-

gebraic geometry. Elimination is the problem of finding the defining ideal of the image

of an algebraic variety under a projection onto a coordinate subspace. Implicitization is

the problem of transforming a given parametric representation of an algebraic variety into

its implicit representation as the zero set of polynomials. Most algorithms for elimina-

tion and implicitization are based on multivariate resultants or Grobner bases, but current

implementations of these algorithms tend to be too slow to handle large instances.

The tropical approach to these problems is based on the following idea. Rather

than computing the variety V (I) of an ideal I by algebraic means, we shall compute the

tropical variety T (I) ⊂ Rn by combinatorial means. Here we provide an alternative to get

some information about the unknown ideals when Grobner bases and resultants fail.

Our setup for elimination problems is the following. Given a closed subvariety

X ⊂ (C∗)n and a morphism of tori α : (C∗)n → (C∗)k, our objective is to compute the

72

Zariski closure of the image of X under α. This is a closed subvariety of (C∗)k which denote

by α(X). In concrete instances the morphism α is represented by a k × n integer matrix

A = (aij), and the subvariety X is given by its ideal IX in the Laurent polynomial ring

C[x±] = C[x±11 , . . . , x±1

n ]. The desired elimination ideal Iα(X) is the inverse image of the

given ideal IX under the C-algebra homomorphism

α∗ : C[u±] 7→ C[u±11 , . . . , u±1

k ] → C[x±11 , . . . , x±1

n ], ui 7→n∏

j=1

xaij

j .

Let f1, . . . , fn ∈ C[t±11 , . . . , t±1

d ] be Laurent polynomials with Newton polytopes

P1, . . . , Pn ⊂ Rd and with supports Ai ⊂ Pi. We assume that each fi is generic relative to

its support, which means that the coefficient vector of fi lies in a Zariski open subset of CAi

where Ai ⊂ Pi is the support of fi. The aim of implicitization is to compute the prime ideal

I of all polynomials g ∈ C[x1, . . . , xn] which satisfies g(f1, . . . , fn) ≡ 0 in C[t±11 , . . . , t±1

d ]. We

here seek to read off as much information as possible about the variety V (I) from P1, . . . , Pn.

Implicitization is a special case of elimination where X is the graph of the parametrization

and α is the projection onto the image coordinates. The desired prime ideal I is the defining

ideal Iα(X) of the image α(X).

Of particular interest is the case when n = d + 1 and I = 〈 g 〉 is a principal ideal.

Here the problem is to predict the Newton polytope Q of the hypersurface V (I) = {g = 0}

from P1, . . . , Pn. This problem was posed in [SY94] and has remained open for over a

decade. It reappeared in recent work of Emiris and Kotsireas [EK05]. We here present a

general solution to the problem of [SY94], namely, a construction of the Newton polytope

Q in terms of the input polytopes Pi. When the variety V (I) is not a hypersurface but has

codimension greater than one, the role of Q is played by the Chow polytope [KSZ92], and

73

our construction generalizes to that case.

The motivation behind both [EK05] and [SY94] is that a priori knowledge of the

Newton polytope Q would greatly facilitate the subsequent computation of recovering the

coefficients of g(x) from the coeffcients of f1(t), . . . , fn(t). This is a problem of linear algebra,

and it will be discussed in Subsection 4.4.1. Part of this chapter is based on a joint work

with Bernd Sturmfels and Jenia Tevelev [STY06].

4.1 The tropical variety of a generic complete intersection

Let IX = 〈f1, . . . , fd〉 ⊂ C[x1, . . . , xn] be a generic complete intersection, i.e. the va-

riety V (IX) has dimension n−d and the coefficients of f1, . . . , fd are generic. Let P1, . . . , Pd

be the Newton polytopes of f1, . . . , fd, and F1, . . . , Fd be their normal fans. Let F be

the common refinement of F1, . . . ,Fd in Rn. It is the normal fan of the Minkowski sum

P = P1+· · ·+Pd. For w ∈ Rn, let facew(Pi) denotes the face of the polytope Pi at which the

linear functional u 7→ u · w attains its minimum. The collection of facew(Pi), i = 1, . . . , d,

depends only on which cone of F contains w in its relative interior.

Let Γ be an n − d-dimensional cone in F and w ∈ Γ be a point in its relative

interior. Then Γ is contained in a cone of dimension at least n − d in each normal fan

Fi, so the polytopes facew(Pi) have dimensions at most d and lie in a d-dimensional affine

subspace orthogonal to Γ. Hence it makes sense to talk about their mixed volume.

The following theorem allows us to compute the tropical variety T (IX) by doing

only polyhedral computations.

Theorem 4.1. The tropical variety T (IX) is the union of n− d-dimensional cones Γ in F

74

such that the mixed volume MV (faceΓ(P1), . . . , faceΓ(Pd)) is positive. The multiplicity of Γ

is equal to this mixed volume.

Consider the system of equations

inw(f1)(t) = · · · = inw(fd)(t) = 0. (4.1)

The Newton polytopes of these Laurent polynomials are facew(P1), . . . , facew(Pd). Let s

denote the dimension of the polytope

facew(P ) = facew(P1) + · · ·+ facew(Pd). (4.2)

Notice that s ≤ d because all the facew(Pi) lie in parallel d-dimensional affine spaces. The

equations (4.1) can be rewritten as a system of Laurent polynomials in s unknowns. We

wish to determine whether or not the system (4.1) has a solution t in the algebraic torus

(C∗)n. Since the coefficients of the fi are assumed to be generic, a necessary condition for

(4.1) to be solvable is that s = d. Otherwise, we have d polynomials with generic coefficients

in fewer than d variables, which have no solution. By Bernstein’s Theorem [Ber75, Kho99],

the number of solutions to the system (4.1) equals

MixedVolume(

facew(P1), . . . , facew(Pd))

, (4.3)

where the mixed volume is normalized with respect to the lattice parallel to the affine span

of facew(P ). In degenerate cases this mixed volume may be zero.

The following lemma is an important step in the proof of Theorem 4.1.

Proposition 4.2. For any w ∈ Rn the following are equivalent:

1. The vector w lies in T (IX).

75

2. The ideal 〈 inw(fi) : i = 1, . . . , d 〉 contains no monomial.

3. The initial system (4.1) has at least one solution t ∈ (C∗)n.

4. The mixed volume (4.3) is positive.

5. For all subsets K ⊆ {1, . . . , d}, the polytope facew(∑

i∈K Pi) has dimension at least |K|.

Proof. Since monomials are units in the Laurent polynomial ring, Hilbert’s Nullstellensatz

shows that (2) and (3) are equivalent. The equivalence of (3) and (4) follows from Bernstein’s

Theorem [Ber75]. The equivalence of (4) and (5) appears in [Ewa96, Theorem 4.13].

Now (1) holds if and only if there is no monomial in inw(IX) . This condition

implies (2) but, a priori, it may be stronger. To see that they are equivalent, we will use

the hypothesis that the coefficients of the fi are generic and show that (3) implies (1).

Suppose that (3) holds, and let t0 ∈ (C∗)d be a solution of the system of equations (4.1).

Using the polyhedral homotopy of [HS95], we can construct a solution t1 of the equations

f1(t) = · · · = fd(t) = 0 which degenerates to t0 under the one-parameter torus given by w.

The deformation t0 → t1 ensures that t0 is a point in the variety of the initial ideal inw(IX),

which therefore contains no monomial.

Proof of Theorem 4.1. Since IX is a complete intersection, by the Unmixedness Theorem

[Eis95, Corollary 18.14], all minimal primes of IX have codimension d, and every associated

prime is a minimal prime. Since the tropical variety T (IX) is pure of dimension n− d, the

Proposition 4.2 above implies the first statement of the theorem. Recall that the multiplicity

of a maximal cone Γ in T (IX) is the sum of multiplicities of all monomial-free minimal

associated primes of inw(IX), where w is any vector in the relative interior of Γ. This is the

76

number of solutions, counted with multiplicities, of the initial system (4.1) after a monomial

change of coordinate so that the ideal becomes zero-dimensional. By Proposition 4.2 agin,

this multiplicity equals the mixed volume (4.3).

4.2 Tropical elimination and implicitization

In the tropical setting, the problem of computing the image α(X) of the given

variety X is replaced by the problem of computing the tropical variety of the image, along

with multiplicities for the maximal cones.

The tropicalization of the image of X under the morphism α is the tropical variety

obtained by taking the image of the fan T (IX) under the linear map A:

T (Iα(X)) = A · T (IX). (4.4)

This formula follows from [Tev, Remark 3.2]. The main problem we are facing is to compute

the correct multiplicities on this fan.

In [ST], Sturmfels and Tevelev show the following formula for the push-forward

of multiplicities. We will assume that the morphism α is generically finite of degree δ. We

also put a fine enough fan structure on T (Iα(X)) such that the image A(Γ) of any cone Γ

in T (IX) is a union of cones in T (Iα(X)). Then the multiplicity of a maximal cone Π in

T (Iα(X)) is

mΠ =1

δ

∑

Γ:A(Γ)⊇Π

mΓ · index(Γ,Π),

where mΓ is the multiplicity of Γ in T (IX), and index(Γ,Π) is the index of the sublattice

generated by A(Γ ∩ Zk) inside the sublattice generated by Π ∩ Zk.

77

We now turn our attention to implicitization, which will be our main application.

Our implicitization problem is specified by a collection of n Laurent polynomials

fi(t) =∑

a∈Ai

ci,a · ta11 · · · t

ad

d (i = 1, 2, . . . , n). (4.5)

Here each Ai is a finite subset of Zd, and the ci,a are generic complex numbers. Our ultimate

aim is to compute the ideal I ⊂ C[x1, . . . , xn] of algebraic relations among f1(t), . . . , fn(t),

or at least, some information about its variety V (I).

To phrase the implicitization problem as an elimination problem, we will consider

the graph G ⊂ (C∗)d × Cn of the morphism f : (C∗)d → Cn defined by f : (t1, . . . , td) 7→

(f1(t), . . . , fn(t)), i.e. the graph G is the closure of {(t, x) : x = f(t)} in (C∗)d ×Cn. Hence

the defining ideal G is

IG = 〈xi − fi(t1, . . . , td) : i = 1, . . . , n〉. (4.6)

Our parametrized variety is the closure of the image of f , which is the same as the projection

of the graph G on the last n coordinates.

Let Ψ : Rd → Rn be the tropicalization of the map f = (f1, . . . , fn), i.e., let

Ψi(w) = min{w ·v : v ∈ Pi} be the support function of the Newton polytope Pi = conv(Ai).

The image of Ψ is contained in the tropical variety T (I), by [PS04, Theorem 2], but this

containment is usually strict. In other words, the image of the tropicalization of f is usually

a proper subset of the tropicalization of the image of f . The point of the following result is

to characterize the difference T (I)\image(Ψ).

Let e1, . . . , en be the standard basis of Rn. For J ⊆ {1, . . . , n}, we abbreviate the

orthant R≥0{ej : j ∈ J} by RJ≥0 and the Minkowski sum

∑

j∈J Pj by PJ .

78

Theorem 4.3. Let f1, . . . , fn be Laurent polynomials in d variables which are generic rel-

ative to their support, and let I ⊂ C[x1, . . . , xn] be the prime ideal of all algebraic relations

among f1, . . . , fn. Then the following subsets of Rn coincide:

1. the tropical variety T (I),

2. the union of all sets Ψ(T (〈fj : j ∈ J〉)) + RJ≥0, where J ⊆ {1, 2, . . . , n},

3. the union of the cones Ψ(w) + RJ≥0, where w ∈ Rd and J runs over all subsets of

{1, 2, . . . , n} such that 〈 inw(fj) : j ∈ J 〉 contains no monomial,

4. the union of the cones Ψ(w) + RJ≥0 such that, for all subsets K ⊆ J , the linear form

w attains its minimum over PK at a face of dimension at least |K|.

Characterization (4) gives us a combinatorial algorithm for computing the tropical

variety T (I) directly from the given Newton polytopes P1, . . . , Pn.

It is instructive to note that the contribution of the empty set J = ∅ in Theorem

4.3 (2) is precisely the image of the tropicalization Ψ of the given map f :

Ψ(T (〈∅〉)) + R∅≥0 = Ψ(T ({0})) = Ψ(Rd) = image(Ψ). (4.7)

Thus it is the contributions made by non-empty subsets J which make up the difference

between the tropicalization of the image and the image of the tropicalization.

Proof of Theorem 4.3. The Theorem 4.1 gives us a combinatorial description of the tropical

variety T (IG) of the graph G defined by ideal IG in (4.6). The Newton polytopes of the

generators in (4.6) are Pi = conv(Pi ∪ (0, ei)) ⊂ Rd+n for i = 1, . . . , n, i.e. each Pi is

a pyramid over Pi with apex (0, ei). Here we view Pi as lying in the first d-coordinate

subspace in Rd+n, and e1, . . . , en are standard unit vectors in the last n coordinates.

79

We now describe the rays of the normal fan F of the Minkowski sum P = P1 +

· · ·+ Pn. The vector (0, ei) ∈ Rd+n is a ray in F if

face(0,ei)(P ) = P1 + · · ·+ Pi + · · ·+ Pn

is a facet. Suppose a vector (w, z) ∈ Rd+n with w 6= 0 is a ray in F . Then we must have

z = Ψ(w). Otherwise, face(w,z)(P ) is strictly contained in face(w,Ψ(w))(P ). The tropical

variety T (IG) is d-dimensional, and the d-dimensional cones in F have the form {(w,Ψ(w)) :

w ∈ C} + (0, RJ≥0) ⊂ Rd+n where C is a d − |J | dimensional cone in the normal fan of

P1 + · · · + Pn. The image of this cone under the projection onto the last n coordinates

is Ψ(C) + RJ≥0. Note that face(C,J)(Pi) is equal to conv(faceC(Pi) ∪ (0, ei)) if i /∈ J and

conv(faceC(Pi)) if i ∈ J . Now we have to determine which pairs (C, J) actually give a

d-dimensional cone in the tropical variety T (IG). Using the equivalence of (1) and (5) in

Proposition 4.2, these are the pairs (C, J) where

dim

(

∑

i∈K

face(C,J)(Pi)

)

≥ |K| for all K ⊂ {1, . . . , d}. (4.8)

We saw above that

dim

(

∑

i∈K

face(C,J)(Pi)

)

= |K\J |+ dim

(

∑

i∈K

faceC(Pi)

)

.

Hence the condition (4.8) is equivalent to

dim

(

∑

i∈K

faceC(Pi)

)

≥ |K ∩ J | for all K ⊂ {1, . . . , d}. (4.9)

However, in condition (4.9) it suffices to check for K ⊂ J . This proves the equivalence of

(1) and (4). The equivalence of (2),(3), and (4) follows from Proposition 4.2.

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We now consider the prime ideal I of algebraic relations among the generic Laurent

polynomials f1(t), . . . , fn(t). Generators for the ideal I are (still) unknown, but in Theorem

4.3 we can compute its tropical variety T (I) in combinatorial terms. In what follows we

similarly compute the multiplicity MΓ for every maximal cone Γ in a fan structure on T (I).

The image of the cones in T (IX) under A do not generally form a fan, and they

need to be subdivided to give a fan structure on the tropical variety T (Iα(X)). We choose a

fine enough fan structure on T (Iα(X)) so that the image of any cone in T (IX) is the union

of some cones of T (Iα(X)). We say that a pair (C, J) covers a d-dimensional cone Γ in

T (Iα(X)) if C is d-dimensional and contains Γ. There may be more than one pair (C, J)

which covers a fixed cone Γ. Suppose that (C, J) covers Γ. Then the sublattice generated by

Ψ(C ∩Zd) + ZJ has rank d in Zn. We define index(C, J) to be the index of that sublattice

in the maximal rank d sublattice of Zn that contains it.

Let Fj = facew(Pj) and FJ = facew(∑

j∈J Pj) =∑

j∈J Fj for some (hence any)

vector w in the relative interior of C. Then the |J |-dimensional mixed volume

MixedVolume(Fj : j ∈ J) (4.10)

is the same as the one in equation (4.3). This mixed volume is normalized with respect

to the affine lattice spanned by the |J |-dimensional polytope FJ . Multiplying (4.10) by

index(C, J) we obtain the scaled mixed volume

MixedVolume(Fj : j ∈ J) · index(C, J) (4.11)

The following theorem follows from the push forward of multiplicities formula seen above.

Theorem 4.4. The multiplicity MΓ of a maximal cone Γ in the fan structure on T (I) is

the sum of all scaled mixed volumes (4.11) where the pair (C, J) covers Γ.

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4.2.1 Examples

We now present some families of examples which illustrate Theorems 4.3 and 4.4.

Let us begin by examining the simplest case: a parametrized curve in the plane.

Example 4.5 (Plane curves). Let d = 1 and n = 2. We wish to compute the equation

g(x1, x2) of the plane algebraic curve parametrized by two Laurent polynomials x1 = f1(t)

and x2 = f2(t). The Newton polytopes of f1 and f2 are segments on the line R1,

P1 = [ a , b ] and P2 = [ c , d ]. (4.12)

Here a ≤ b and c ≤ d are integers.

Figure 4.1: Tropical plane curves and their Newton polygons

Now let’s see how we get the tropical variety and the multiplicities. For J = {1, 2}

we just get the empty set since 〈f1, f2〉 is the unit ideal in C[t, t−1]. For J = {1} we get

the ray spanned by (1, 0) since T (〈f1〉) = {0}. Its multiplicity is the the volume of the first

Newton polytope, which is equal to b− a. Similarly for J = {2} we get the ray spanned by

82

(0, 1) with multiplicity d − c. Finally, for J = ∅ we get the image of the tropicalization Ψ

which consists of the two rays spanned by (a, c) and by (−b,−d). See Figure 4.1.

The Newton polygon Q of the implicit equation g(x1, x2) is a (possibly degenerate)

quadrangle in R2. There are four cases:

• if a ≥ 0 and c ≥ 0 then Q = conv{

(0, b), (0, a), (c, 0), (d, 0)}

,

• if b ≤ 0 and d ≤ 0 then Q = conv{

(0,−a), (0,−b), (−d, 0), (−c, 0)}

,

• if a ≤ 0, d ≥ 0 and bc ≥ ad then Q = conv{

(0, b − a), (0, 0), (d − c, 0), (d,−a)}

,

• if b ≥ 0, c ≤ 0 and bc ≤ ad then Q = conv{

(0, b− a), (0, 0), (d − c, 0), (−c, b)}

.

The possible quadrangles Q and their normal fans are illustrated in Figure 4.1. We note

that the case a = 48, b = 63, c = d = 32 appears in [EK05, Example 3.3].

Monomials

Suppose that each fi is a monomial, so each Pi is just a point, say Pi = {ai} ⊂ Zd.

Then I is the toric ideal generated by all binomials xu − xv where u− v is in the kernel of

the matrix A = [a1, a2, . . . , an]. The tropical variety T (I) equals the row space of A. The

tropical morphism Ψ : Rd → Rn is the linear map w 7→ wA given by left multiplication

with A. For any nonempty subset J of {1, 2, . . . , n}, the ideal 〈fj : j ∈ J〉 is the unit ideal,

which contributes nothing to the union in (2) of Theorem 4.3. The ideal corresponding to

J = ∅ is 〈∅〉 = {0}, and T (I) = rowspace(A) is indeed the image of T (〈∅〉) = Rd under

the map Ψ.

83

The Unmixed Case

Suppose that all Newton polytopes are dilations of a common polytope, say Pi =

ciP for some positive integers ci and a full-dimensional polytope P ⊂ Rd. We distinguish

two cases.

First suppose that P contains the origin. Then the image of Ψ is the halfline

C− = R≤0(c1, . . . , cn). In fact, the set Ψ(T (〈fj : j ∈ J〉)) in Theorem 4.3 (2) equals C−

if |J | < d, it equals {0} if |J | = d, and it is empty if |J | > d. Therefore, the tropical

variety T (I) is the union of the cones RJ≥0 for J ⊂ {1, . . . , n} with |J | = d and the cones

C− + RJ≥0 for J ⊂ {1, . . . , n} with |J | = d− 1.

Next suppose that P does not contain the origin. Then the image of Ψ is the line

C = R(c1, . . . , cn), and we conclude that the tropical variety T (I) is the union of the cones

RJ≥0 for |J | = d, and the cones C + RJ

≥0 for |J | = d− 1.

In the case n = d + 1, the tropical variety T (I) just constructed has codimension

one in Rn. It is the union of all codimension one cones in the normal fan of the Newton poly-

tope Q of the implicit equation. The polytope Q was constructed with a different technique

in [SY94, Theorem 9], and the previous two paragraphs are consistent with that result.

In the first case when 0 ∈ P , the polytope Q is a simplex, which is the convex

hull of 0,m1e1,m2e2, . . . ,md+1ed+1. Each mi is equal the length of the edge from 0 to

miei, hence the multiplicity of its normal cone spanned by {ej : j 6= i}. This is the mixed

volume of the corresponding d polytopes {Pj : j 6= i} in Rd. The total degree of the implicit

equation g is the maximum of the total degrees of the extreme monomials, which correspond

to vertices of the Newton polytope Q, so the degree of g is the maximum of m1, . . . ,md+1.

84

Example 4.6 (Full support). An interesting special case is when the input polynomials

f1, . . . , fd+1 have full support of degrees a1, . . . , ad+1 respectively. Then the Newton poly-

topes are ai∆d wehre ∆d is the standard simplex conv{0, e1, . . . , ed}. In that case, the mixed

volume of {Pj : j 6= i} is mi =∏

j 6=i aj by the Bernstein and Bezout theorems. As shown

above, the Newton polytope of the implicit equation g is conv(0,m1e1, . . . ,md+1ed+1).

Linear maps

Suppose each support Ai is a subset of {e1, . . . , ed}, the unit vectors in Rd. Thus

fi is a generic linear form in the set of unknowns { tj : ej ∈ Ai}. The ideal I is generated

by an (n − d)-dimensional space of linear forms in C[x1, . . . , xn]. We introduce the subset

Bj = {i : ej ∈ Ai} for j = 1, 2, . . . , d. The rank d matroid corresponding to the linear space

V (I) ⊂ Cn is the transversal matroid [Bru87] of the set family {B1, . . . , Bd}. The tropical

variety T (I) is the Bergman complex of that transversal matroid, as shown in [AK06].

Thus, in the linear case, Theorem 4.3 offers an interesting new representation of

the Bergman fans of transversal matroids. It can be described as follows. The sum of

simplices P =∑n

i=1 conv(Ai) is a generalized permutohedron [Pos], which means that the

normal fan of P is a coarsening of the Sn-arrangement. Let C be any cone in that fan. Each

initial form inC(fj) is supported on a subset faceC(Aj) of Aj. Then with C we associate

the simplicial complex ∆C on {1, 2, . . . , n} whose simplices are the subsets J such that these

initial forms have a solution in (C∗)d. This is a combinatorial condition on the set family

{ faceC(Aj) : j ∈ J }, which is essentially Postnikov’s Dragon Marriage Condition [Pos, §5].

To construct the Bergman fan, we map the normal fan of P from Rd into Rn using the map

Ψ, and to each image cone Ψ(C) we attach the family of orthants RJ≥0 indexed by ∆C .

85

Example 4.7. Let d = 3, n = 6 and consider six linear forms

f1 = c11 · t1 + c12 · t2

f2 = c21 · t1

f3 = c31 · t1 + c33 · t3

f4 = c42 · t2

f5 = c53 · t3

f6 = c62 · t2 + c63 · t3,

where the cij are general complex numbers. The tropical linear space T (I) is three-

dimensional in R6, but each maximal cone contains the line spanned by (1, 1, 1, 1, 1, 1),

so we can represent T (I) by a graph. This graph is the Bergman complex of a rank 3

matroid on {1, 2, 3, 4, 5, 6}, namely, the transversal matroid of

(B1, B2, B3) =(

{1, 2, 3}, {1, 4, 6}, {3, 5, 6})

.

6

��

��

��

��

��

��

��

��

456

4

1

32

235

5

124

��

Figure 4.2: The Bergman complex of a transversal matroid

The drawing of T (I) in Figure 4.2 is the same as [DFS, Figure 2], but we now

86

derive it from Theorem 4.3 (2). The polytope P = P1 + · · ·+P6 is a planar hexagon, and Ψ

maps the six two-dimensional cones in its normal fan to the six edges in the outer hexagon in

Figure 4.2. Each of the edges {124, 1}, {2, 6}, {235, 3}, {5, 1}, {456, 6}, and {4, 3} is between

the image of a ray in the normal fan of P and a ray generated by e1, e3, or e6. For instance,

the edge {124, 1} corresponds to the ray generated by Ψ ((1, 1, 0)) = (1, 1, 0, 1, 0, 0) plus the

ray R≥0 e1 in R6. None of the simplicial complexes ∆C contains 2, 4, or 5 because f2, f4,

and f5 are monomials. However, e2, e4, and e5 are images of some rays in the normal fan

of P under the tropical map Ψ. The triangle in the middle of Figure 4.2 represents the

simplicial complex associated with the 0-dimensional face {0} of the normal fan:

∆{0} ={

{1, 3}, {1, 6}, {3, 6}}

.

This shows how tropical implicitization works for the linear forms f1, . . . , f6.

Binomials

Consider the case where the map f is given by n binomials

fi = ci1 · tai + ci2 · t

bi (i = 1, 2, . . . , n).

Each Newton polytope Pi = conv(ai, bi) is a line segment, and their Minkowski sum P =

P1 + · · · + Pn is a zonotope (i.e. a projection of the n-cube). The normal fan of P is the

hyperplane arrangement H ={

{u · aj = u · bj}}

j=1,...,n. The map Ψ : Rd → Rn is the

tropical morphism associated with the arrangement H, that is,

Ψ(u) =(

min(u · a1, u · b1),min(u · a2, u · b2), . . . ,min(u · an, u · bn))

.

This map was recently studied by Ardila [Ard] for certain graphic arrangements H.

87

Theorem 4.3 shows that the tropical variety T (I) is the union of the cones Ψ(C)+

RJ≥0, where C is any cone of the hyperplane arrangement H, and the hyperplanes indexed

by J ⊂ {1, . . . , n} contain the cone C and are linearly independent.

Example 4.8. Let d = 2 and consider the following n = 3 binomials in t1 and t2:

f1 = (t1 − α) · tu11 tu2

2 , f2 = (t2 − β) · tv11 tv2

2 , f3 = (t1 − γt2) · tw11 tw2

2 ,

where α, β, γ are general complex numbers, and u1, u2, v1, v2, w1, w2 are integers. Then H

is the arrangement of three lines in the plane R2 which consists of the two coordinate axes

and the main diagonal. The images of the six one-dimensional cones of the arrangement H

under the map Ψ are spanned by the vectors

Ψ(e1) = (u1, v1, w1),

Ψ(e1 + e2) = (u1 + u2, v1 + v2, w1 + w2 + 1),

Ψ(e2) = (u2, v2, w2),

Ψ(−e1) = (−u1 − 1,−v1,−w1 − 1),

Ψ(−e1 − e2) = (−u1 − u2 − 1,−v1 − v2 − 1, w1 − w2 − 1),

Ψ(−e2) = (−u2,−v2 − 1,−w2 − 1).

The image of Ψ consists of the six two-dimensional cones in R3 which are spanned by

(cyclically) consecutive vectors in this list. Next, the tropical surface T (I) contains the six

two-dimensional cones which are spanned by the following pairs in R3:

{Ψ(e1), e2} , {Ψ(e1 + e2), e3} , {Ψ(e2), e1} ,

{Ψ(−e1), e2} , {Ψ(−e1 − e2), e3} , {Ψ(−e2), e1}.

88

Finally, the zero-dimensional cone C = {0} contributes the three two-dimensional cones

which are spanned by the pairs of standard basis vectors

{e1, e2} , {e1, e3} , {e2, e3}.

These pairs determine a non-planar graph with 9 vertices and 15 edges. It is isomorphic to

the one depicted in Figure 4.2. Thus T (I) is a two-dimensional fan in R3 whose intersection

with the unit sphere is an immersion of that graph. The resulting embedded graph, which

is dual to the Newton polytope of the implicit equation, depends on the numerical values

of the exponents u1, u2, v1, v2, w1, w2. This example shows how our method can be applied

to a family of implicitization problems where the exponents of the fi are not fixed integers

but are unknowns.

Surfaces

Suppose that d = 2, so our given input is a list of lattice polygons P1, . . . , Pn in

R2. The tropical variety T (I) is a two-dimensional fan in Rn which we represent by an

embedded graph in the (n − 1)-sphere Sn−1. This graph is constructed as follows. We fix

the node ei for each polygon Pi which is not just a point, and we fix the node Ψ(w) for each

vector w that is an inner normal to an edge of the polygon P = P1 + · · ·+Pn and such that

Ψ(w) 6= 0. We identify such a vector Ψ(w) with the point Ψ(w)/||Ψ(w)|| on Sn−1.

We now connect pairs of nodes by great circles in Sn−1 according to the follow-

ing rules: a pair {Ψ(w),Ψ(w′)} gets connected if their edges on P are adjacent, a pair

{Ψ(w), ei} gets connected if w is the inner normal to an edge of Pi, and a pair {ei, ej}

gets connected if Pi + Pj is two-dimensional. Some pairs of great circles intersect and thus

89

create new nodes in Sn−1. This intersection need not be transversal, i.e., two great circles

may intersect in a smaller great circle. The result of this construction is the tropical surface

T (I), represented by a graph in Sn−1.

The edges of this graph are weighted with multiplicities as follows. The weight of

a pair {Ψ(w),Ψ(w′)} is the greatest common divisor of the 2×2-minors of the n×2-matrix

(Ψ(w),Ψ(w′)), divided by the determinant of the 2×2-matrix (w,w′). The weight of a pair

{Ψ(w), ei} is the normalized length of the edge facew(Pi) of Pi times the greatest common

divisor of all coordinates of Ψ(w) but the ith. The weight of a pair {ei, ej} is the mixed area

MixedVolume(Pi, Pj) = area(Pi + Pj)− area(Pi)− area(Pj).

Now, when forming the embedded graph T (I) ⊂ Sn−1, each edge Γ may be covered by more

than one of the great circles created by these pairs, and we take MΓ to be the sum of their

weights. With these multiplicities, the graph T (I) is balanced.

Example 4.9. Consider the family of surfaces in 3-space which is given by

A1 = {(1, 0), (3, 1), (2, 2)}, A2 = {(−1, 0), (0,−1), (0, 0)}, A3 = {(2, 1), (0, 2), (1, 3)}.

To construct the graph T (I) on the sphere S2, we first draw the three nodes e1, e2, e3

and the nine nodes Ψ(w) which are the images under Ψ of the inner normals of the 9-gon

P1 + P2 + P3. These nine directions are given by the columns of the matrix

e2 e3 e1 e2 e3 e1 e2 e3 e1

1 1 −1 −3 −7 −4 −2 0 2

−1 −2 −2 0 0 0 0 −1 −2

2 4 0 −2 −5 −4 −3 −2 −2

90

We connect these twelve nodes with great circles as described above, namely, by forming

the 9-cycle of these columns, by connecting them to the ei as indicated, and by forming the

triangle e1, e2, e3. This creates an embedded graph in S2 which has 14 vertices, 27 edges

and 15 regions. The Newton polytope Q of the implicit equation is dual to this graph, so

it has 15 vertices, 27 edges and 14 facets. The cones {e1, e2}, {e1, e3}, and {e2, e3} have

multiplicities 2, 6, and 3 respectively. These are the corresponding mixed volumes of the

input Newton polytopes. All the other cones have multiplicity 1. Using the methods to

be described in Sections 4.3 and 4.4, we construct Q metrically and find that the largest

coordinate sum of a vertex is 14 and that there are 155 lattice points in it. Hence the

implicit equation g(x1, x2, x3) is a polynomial of degree 14 having at most 155 terms.

Here is a concrete numerical example taken from the computer algebra literature.

3

��

��

��

��

��

��

��

��

��

��

��

��

e

ee

3

1 2

5

6

7

4 8 2

��

��

��

��

��

��

��

��

e

48

57

2

ee

6

1 2

Figure 4.3: Tropical construction of a bicubic surface in 3-space

Example 4.10. Let d = 2 and consider the well-known problem of implicitizing bicubic

91

surfaces. The following specific surface was discussed in [EK05, Example 3.4]:

f1(s, t) = 3t3 − 6t2 + 3t + s3 − 3s2 + 6s− 1,

f2(s, t) = 3s3 − 6s2 + 3s + t3 + 3t,

f3(s, t) = −3s3t3 + 15s2t3 − 15st3 − 3s3t2 − 18s2t2 + 27st2

−3t2 + 6s3t + 9s2t− 18st + 3t− 3s2 + 3s.

The vertices of the Newton polygons are indicated in boldface font. The polygon P1 is a

triangle, P2 is a quadrangle, and P3 is a 7-gon. Their Minkowski sum P1 + P2 + P3 is an

octagon. The eight primitive inner normal vectors w = (w1, w2) of this octagon are listed

in the first two rows of the following matrix:

e1e2e3 e2e3 e1e2e3 e3 e3 e1e2 e3 e2

w1 1 1 0 −1 −1 −1 0 1

w2 0 1 1 1 0 −1 −1 −1

Ψ(w)1 0 0 0 −3 −3 −3 −3 −3

Ψ(w)2 0 1 0 −3 −3 −3 −3 −3

Ψ(w)3 0 1 0 −2 −3 −6 −3 −2

The last three rows contain the coordinates of their images Ψ(w) under the tropical mor-

phism Ψ. We now construct the graph as in 4.2.1. The first and third inner normal vector

is mapped to zero under Ψ, so they do not contribute to the tropical surface T (I). The

remaining six columns 2, 4, 5, 6, 7, 8 and the unit vectors e1, e2, e3 form the following 15 great

92

circles on the sphere S2, as shown in Figure 4.3:

pair index mixed volume on edge

{e1, e2} 1 9 {e1, e2}

{e1, e3} 1 18 {e1, e3}

{e2, e3} 1 17 {e2, e3}

{2, e2} 1 1 {e2, e3}

{2, e3} 1 1 {e2, e3}

{6, e1} 3 3 {6, e1}

{6, e2} 3 3 {6, e2}

{4, e3} 3 1 {6, e3}

{8, e3} 3 1 {6, e3}

{5, e3} 3 2 {6, e3}

{7, e3} 3 2 {6, e3}

{4, 5} 3 1 {6, e3}

{5, 6} 9 1 {6, e3}

{6, 7} 9 1 {6, e3}

{7, 8} 3 1 {6, e3}

There is considerable overlap among these great circles. For instance, the points 4 and 8

coincide, as do the points 5 and 7, and they all lie on the great circle between e3 and 6.

Likewise, the point 2 lies on the great circle between e2 and e3. Thus the embedded graph

T (I) ⊂ S2 is the complete graph K4, and, by adding up the contributions of each index

93

times mixed volume, we get the corresponding weights:

edge Γ {e1, e2} {e1, e3} {e2, e3} {6, e1} {6, e2} {6, e3}

weight MΓ 9 18 18 9 9 18

(4.13)

From this table we can now determine the Newton polytope Q of the implicit

equation g(x1, x2, x3). The general method for this is explained in the next section. In this

example we find that the proposed Newton polytope is the tetrahedron

Q ={

(u1, u2, u3) ∈ R3 : u1, u2, u3 ≥ 0 and u1 + u2 + 2 · u3 ≤ 18}

.

The number of lattice points in Q equals 715. What we can conclude at this point is that

Q would be the Newton polytope of the implicit equation g if f1, f2, f3 were replaced by

polynomials with the same support but with generic coefficients.

However, the coefficients of the specific polynomials f1, f2, f3 we took from [EK05,

Example 3.4] are not generic. For instance, the equations f2(s, t) = f3(s, t) = 0 have only 15

solutions (s, t) ∈ (C∗)2, which is less than the number 17 predicted by the mixed volume.

Yet, it turns out that Q is the correct Newton polytope and all 715 possible monomials

appear with non-zero coefficients in the implicit equation:

g(x1, x2, x3) = 387420489x181 + 387420489x18

2 − 18014398509481984x93 + · · · · · ·

−12777985432959891776936639829x22 + · · · − 3707912273492242256259566313.

We included the x22-term because it has the coefficient of largest absolute value.

94

4.3 From tropical variety to Newton polytope to implicit

equation

In the previous sections we constructed the tropical variety T (I) along with the

multiplicity MΓ for each maximal cone Γ. Assuming I = 〈g〉 to be a principal ideal, we now

show how this information reveals the Newton polytope Q of g. The polynomial g is then

recovered from Q and the fi’s using numerical linear algebra. If V (I) is not a hypersurface

then the role of Q will be played by the Chow polytope.

Generic hypersurfaces

The hypersurface case, when n = d + 1, has received the most attention in

the computer algebra literature [CGKW01, EK05, SY94]. Suppose we are given d + 1

Laurent polynomials f0, f1, . . . , fd whose coefficients are generic relative to their supports

A0, A1, . . . , Ad ⊂ Zd. Their Newton polytopes are denoted by P0, P1, . . . , Pd. Note that we

shifted indices by one. The prime ideal of algebraic relations among the fi is denoted, as

before, by I ⊂ C[x0, x1, . . . , xd]. There is a simple combinatorial criterion for when the

ideal I is principal, i.e., when the variety V (I) parametrized by (f0, f1, . . . , fd) is actually

a hypersurface in Cd+1.

Proposition 4.11. The prime ideal I is principal if and only if there exist points ai ∈ Ai

such that the d× (d + 1)-matrix (a0, a1, . . . , ad) has maximal rank d.

Proof. This can be seen by writing down the d × (d + 1)-Jacobian matrix(

∂fi/∂tj)

, and

using the fact that the coefficients of the fi are generic.

95

We now assume that the condition in Proposition 4.11 is satisfied, and we let

g(x0, x1, . . . , xd) denote the unique (up to scaling) generator of the ideal I. Let Q ⊂ Rd+1

be the Newton polytope of the irreducible polynomial g. Then the tropical hypersurface

T (I) = T (g) is the union of the codimension one cones in the normal fan of Q, and the

multiplicity of a maximal cone Γ in any fan structure on T (I) equals to lattice length of

the edge of Q whose normal cone contains Γ.

Suppose we constructed the tropical hypersurface T (I) using Theorem 4.3, and

we computed the multiplicities MΓ of each cone Γ in T (I) using Theorem 4.4. These data,

combined with the requirement that Q lies in the non-negative orthant and intersects each

of the d + 1 coordinate hyperplanes, determine the polytope Q uniquely. Hence we can

construct the desired Newton polytope Q ⊂ Rd+1 combinatorially from the given Newton

polytopes P0, P1, . . . , Pd ⊂ Rd.

The previous paragraph almost solves the problem stated in [SY94]. The only

shortcoming is that the description of the set T (I) given in Theorem 4.3 does not come

with a nice fan structure as we will see in Examples 4.9 and 4.10. In our view, the following

theorem provides a better solution, as it does not require the knowledge of any fan structure.

All that is needed are the integers Mγ = MΓ > 0 where γ is a smooth point on T (I) and

Γ is any sufficiently small relatively open cone in T (I) which contains γ. These numbers

Mγ are computed using the formula in Theorem 4.4, simply by replacing “(C, J) covers Γ”

with “(C, J) covers γ”.

The following theorem gives a formula for each coordinate of each vertex of the

Newton polytope Q. It does not rely on any particular fan structure on T (I).

96

Theorem 4.12. For a generic vector v ∈ Rd+1, the ith coordinate of the vertex facev(Q)

equals the number of intersection points, each counted with its intersection multiplicity, of

the tropical hypersurface T (I) with the halfline v + R≥0 ei.

This result and its generalization to larger codimension, to be stated in Theorem

4.15 below, is due to Dickenstein, Feichtner, and Sturmfels [DFS]. We consider the con-

junction of Theorems 4.3, 4.4 and 4.12 to be a satisfactory solution to the problem stated

in [SY94].

What remains is for us to explain the meaning of the term “intersection multiplic-

ity” in the statement of Theorem 4.12. Since v is generic, each intersection of the tropical

hypersurface T (I) with the halfline v + R≥0ei occurs in a smooth point γ of T (I). The

intersection multiplicity at γ is the product of the multiplicity Mγ and the index of the

finite index sublattice Zei + (RΓ ∩ Zd+1) of Zd+1. In [DFS, §2], these two factors were

called the intrinsic and extrinsic multiplicities of the intersection.

Non-generic hypersurfaces

The following proposition ensures that we can apply our tropical method even if

the given polynomials fi(t) do not have generic coefficients. If there is a unique irreducible

relation among the given fi, then our linear algebra method will find that relation, up to a

polynomial multiple.

Proposition 4.13. Let f0, . . . , fd ∈ C[t±11 , . . . , t±1

d ] be any Laurent polynomials whose ideal

of algebraic relations is principal, say I = 〈g〉, and Pi ⊂ Rd the Newton polytope of fi. Then

the polytope Q, which is constructed combinatorially from P0, . . . , Pd as in Subsection 4.3,

97

contains a translate of the Newton polytope of g.

Proof. We introduce a family of Laurent polynomials f ǫ0(t), f

ǫ1(t), . . . , f

ǫd(t) with coefficients

in C(ǫ) which are generic for their support and which satisfy limǫ→0fǫi (t) = fi(t) for all

i. There exists an irreducible polynomial gǫ(x0, x1, . . . , xd) with coefficients in C(ǫ) which

vanishes if we replace each xi by f ǫ(t). Removing common factors of ǫ from the terms of

gǫ, we may assume that

g(x) := limǫ→0 gǫ(x)

is not the zero polynomial. The Newton polytope of g is contained in Q. Now, we take the

limit for ǫ→ 0 in the identify

gǫ(

f ǫ0(t), f

ǫ1(t), . . . , f

ǫd(t)

)

≡ 0 in C(ǫ)[t±11 , . . . , t±1

d ].

This implies that g(x) is an algebraic relation among f0(t), f1(t), . . . , fd(t), so the irreducible

polynomial g(x) is a factor of g(x). This means that the Newton polytope of g is a Minkowski

summand of the Newton polytope of g. This implies that a translate of the Newton polytope

of g lies in Q.

Example 4.14. Let d = 2 and consider the three homogeneous quadrics:

f0(s, t) = s2 + st− 2t2

f1(s, t) = s2 − 2st + t2

f2(s, t) = −2s2 + st + t2

Here, g = x0 + x1 + x2, so its Newton polytope is the triangle ∆ = conv{e0, e1, e2}. The

Newton polytope of the implicit equation for a parametric surface given by three general

98

quadrics is twice that triangle: Q = 2 ·∆. This shows that ∆ is not a subpolytope of Q but

a (non-unique) translate of ∆ is a subpolytope of Q.

The approach suggested by Proposition 4.13 is to not worry at all whether the

given polynomials have generic coefficients or special coefficients. This will not be relevant

until the very end, when the numerical linear algebra detects that the solution space to the

linear system of equations for the coefficients of g(x) is larger than expected. This should

work well when the fi are not too far from the generic case. However, for the kind of special

morphisms f which typically arise in algebraic geometry and its applications (for instance,

in statistics), this approach is likely to fail in practice.

Lower-dimensional varieties

We now consider the case when the parametric variety V (I) ⊂ Cn and its tropi-

calization T (I) ⊂ Rn do not have codimension one. For technical reasons, we here assume

that the fi are homogeneous polynomials of the same degree, so that I is a homogeneous

prime ideal in C[x1, . . . , xn]. Here the role of the Newton polytope Q is played by the Chow

polytope Chow(I). This is the lattice polytope in Rn which is defined as follows. For a

generic vector v ∈ Rn, the ith coordinate of the vertex facev(Chow(I)) of Chow(I) is the

sum of µP (inv(I)) where P runs over all monomial primes that contain the variable xi and

µP (inv(I)) denotes the multiplicity of the monomial ideal inv(I) along P . The Chow poly-

tope was introduced and studied in [KSZ92]. If V (I) has codimension one, then Chow(I)

is precisely the Newton polytope of the irreducible generator g of I.

The following result from [DFS, Theorem 2.2] shows that in the generic case we

99

can construct Chow(I) from the tropical variety T (I) and its multiplicities Mγ in the same

way as constructing the Newton polytope of the implicit equation of a hypersurface. Let

I be any prime ideal of height c in C[x1, . . . , xn] and v a generic vector in Rn, so that the

initial ideal inv(I) is generated by monomials.

Theorem 4.15 (DFS). A prime ideal P = 〈xi1 , . . . , xic〉 is a minimal prime of inv(I) if

and only if the tropical variety T (I) meets the cone v + R≥0{ei1 , . . . , eic}. The number of

intersections, each counted with its intersection multiplicity, coincides with the multiplicity

µP (inv(I)) of the monomial ideal inv(I) along P .

Geometrically, the vertices of Chow(I) correspond to the toric degenerations of the

algebraic cycle underlying I. All points in Chow(I) have the same coordinate sum, namely,

the degree of the projective variety of I. Thus we get a combinatorial rule for computing

the degree of the image of any projective morphism. Moreover, if I is homogeneous with

respect to some multigrading, then Theorem 4.15 gives us a formula also for the multidegree

of I. The multidegree is a multivariate polynomial which refines the classical notion of

degree, and which has received much attention recently in algebraic combinatorics through

the work of Knutson and Miller [KM05].

4.4 TrIm: Software for tropical implicitization

We present software called TrIm for tropical implicitization. It also computes

the tropical variety of an elimination ideal of a generic complete intersection. A special

case of interest to us is when the elimination ideal is principal. In this case, according to

Khovanskii and Esterov [KE], our computation is equivalent to computing a mixed fiber

100

polytope [McM04].

4.4.1 Algorithms and implementation

We will compute the tropical variety of the implicit ideal by projecting the tropical

variety of the graph. So our first task is to compute the tropical variety of a generic complete

intersection ideal generated by n polynomials with generic coefficients in d + n variables.

The correctness of the following algorithm is the content of Theorem 4.1 above.

Algorithm 4.16 (Tropical variety of a generic complete intersection). Input A set of n

Laurent polynomials in d + n variables.

Output The tropical variety, with multiplicities, of the corresponding generic complete in-

tersection.

Do 1. List the exponent vectors of the input polynomials.

2. Compute the Minkowski sum of the Newton polytopes P1, . . . , Pn, and output the

inner normal vectors of the facets.

3. For each d dimensional cone C in the normal fan,

• Let w be a vector in the interior of C.

• Compute the n-dimensional mixed volume MV (facew(P1), . . . , facew(Pn)).

• If the mixed volume is positive, output the generators of the cone and the

multiplicity, which is the mixed volume times the index of the lattice gen-

erated by points in facew(P1), . . . , facew(Pd) (shifted to the origin) in its

saturation.

101

In our implementation of TrIm, the Minkowski sum is computed using the iB4e

library [Hug06], enumerating the d-dimensional cones is done by polymake [GJ00], the

mixed volumes are computed using the mixed volume library [EC95], and the integer

linear algebra for lattice indices is done using NTL [Sho]. What oversees all the computations

is a perl script. In this case, the output format is consistent with that of the current version

of Gfan [Jen] and is intended to interface with the polyhedral software Polymake [GJ00]

when it supports polyheral complexes and fans in the future.

When n = d + 1, the image variety is hypersurface and the tropical variety is

the normal fan of Newton polytope P of this hypersurface. In this case, TrIm recovers the

polytope P from the tropical variety. The maximal cones and their multiplicities in the

tropical variety gives us the directions and lengths of edges in the polytope. We put them

together using the following algorithm from Theorem 4.12.

Given a generic vector w ∈ Rk such that facew(P ) is a vertex v, the ith coordinate

vi is the number of intersections, counted with multiplicities, of the ray w + R>0ei with the

tropical variety. Here, the multiplicity of the intersection with a cone Γ is the multiplicity

of Γ times the absolute value of the ith coordinate of the primitive normal vector to the

cone Γ.

Intuitively, what we are doing here is the following. We wish to determine the

coordinates of the extreme vertex v = facew(P ) of a polytope P in a given direction w.

Our polytope is placed so that it lies in the positive orthant and touches all the coordinate

hyperplanes. To compute the ith coordinate of the vertex v, we can walk from v toward

102

the ith hyperplane along the edges of P , while keeping track of the edge lengths in the

ith direction. A systematic way to do the walk is to follow the edges whose inner normal

cone intersect the ray w + R>0ei. Recall that the multiplicity of a codim-1 normal cone is

the lattice length of the corresponding edge. Using this subroutine for computing extreme

vertices, we can build the whole polytope using the following algorithm.

Algorithm 4.17 (Beneath/Beyond). Input A codim-1 tropical variety with multiplicities

on the maximal cones.

Output The vertices and facets of the Newton polytope P .

Initialization For some generic vectors w ∈ Rk, compute the vertices facew(P ) and take

their convex hull P ′.

Iteration For each inner facet normal vector w of P ′, compute a vertex v ∈ facew(P ). If

v · w < minx∈P ′ x · w then set P ′ ←conv(P ′ ∪ {v}).

Termination There are no new vertices in any facet normal directions of P ′.

Output Vertices and facets of P = P ′.

The intialization step can be performed systematically so that P ′ is initialized to

be a simplex of dimension dim(P ) by testing O(dim(P )2) choices of w. Our algorithm is

essentially the classical Beneath/Beyond algorithm for computing convex hulls, except that

points are given to us implicitly via a linear programming (LP) oracle rather than a list.

The software iB4e [Hug06] that we used here is designed so that any LP oracle can be

plugged in and then the Beneath/Beyond algorithm is performed with the specified oracle.

103

Example 4.18. Now let us look at how TrIm handles the polynomials in Example 4.10.

The input is the following:

[s,t]

[3*t^3-6*t^2+3*t+s^3-3*s^2+6*s-1,

3*s^3-6*s^2+3*s+t^3+3*t,

-3*s^3*t^3+15*s^2*t^3-15*s*t^3-3*s^3*t^2-18*s^2*t^2+27*s*t^2-3*t^2

+6*s^3*t+9*s^2*t-18*s*t+3*t-3*s^2+3*s]

The tropical variety of the graph of this polynomial map is shown in Table 4.1, in

Gfan / polymake format. The projection onto the last three coordinates is shown in table

4.2. The tropical variety is represented as a union of cones with multiplicities. Notice that

we do not have a nice fan structure for this image. This set of cones is slightly different from

that in Example 4.10. However, the tropical variety as a set and multiplicities at smooth

points are the same. Compare the multiplicites here with those in display formula 4.13.

Table 4.3 shows the Newton polytope of the implicit equation and lattice points in it.

In the hypersurface case, the exponents of monomials with non-zero coefficients in

the implicit equation g all lie in the Newton polytope Q. After computing the vertices of

Q, enumerating the integer points in Q gives us a list of all possible monomials in g with

indeterminate coefficients. Finally, we apply numerical or integer linear algebra to compute

the indeterminate coefficients of the polynomial g.

This is done as follows. Recall that we are given a parametrization (f0, . . . , fd)

of the hypersurface V (I). We can thus pick any point τ ∈ (C∗)d and substitute its image

(x0, . . . , xd) = (f0(τ), . . . , fd(τ)) ∈ V (I) into the equation g(x0, . . . , xd) = 0. This gives us

one linear equation for the coefficients of g. We now pick a different point τ ′ in (C∗)d to get

a second equation, and so on. In this manner, we can generate a system of linear equations

104

AMBIENT_DIM

5

DIM

2

LINEALITY_DIM

0

RAYS

1 -1 -3 -3 -2

1 0 0 0 0

0 0 0 1 0

-1 -1 -3 -3 -6

-1 0 -3 -3 -3

0 0 0 0 1

1 1 0 1 0

0 1 0 0 0

0 0 1 0 0

0 -1 -3 -3 -3

-1 1 -3 -3 -2

MAXIMAL_CONES

0 1

0 5

0 9

1 2

1 5

1 6

1 8

2 3

2 5

2 6

2 7

2 8

3 4

3 8

3 9

4 5

4 10

5 7

5 8

5 9

5 10

6 7

7 8

7 10

MULTIPLICITIES

1

1

1

2

2

1

3

3

18

1

2

9

1

3

1

2

1

2

18

2

1

1

3

1

N_RAYS

11

N_CONES

24

N_INCIDENCES

3

5

6

4

3

8

3

5

5

3

3

Table 4.1: The output of TrIm for the graph in Example 4.18

105

IMAGE_RAYS

-3 -3 -2

0 0 0

0 1 0

-3 -3 -6

-3 -3 -3

0 0 1

0 1 0

0 0 0

1 0 0

-3 -3 -3

-3 -3 -2

IMAGE_CONES

0 1

0 5

0 9

1 2

1 5

1 6

1 8

2 3

2 5

2 6

2 7

2 8

3 4

3 8

3 9

4 5

4 10

5 7

5 8

5 9

5 10

6 7

7 8

7 10

IMAGE_MULTIPLICITIES

0

3

3

0

0

0

0

9

18

0

0

9

9

9

9

6

3

0

18

6

3

0

0

0

Table 4.2: The tropical variety of the implicitization problem in Example 4.18.

106

VERTICES

1 0 18 0

1 18 0 0

1 0 0 9

1 0 0 0

FACETS

0 0 162 0

0 162 0 0

0 0 0 324

2916 -162 -162 -324

VERTICES_IN_FACETS

{1 2 3}

{0 2 3}

{0 1 3}

{0 1 2}

LATTICE_POINTS

1 0 0 0

1 1 0 0

1 2 0 0

1 3 0 0

....

1 1 0 8

1 2 0 8

1 0 1 8

1 1 1 8

1 0 2 8

1 0 0 9

N_LATTICE_POINTS

715

Table 4.3: The output that describe the Newton polytope of the implicit equation and thelattice points in it, for Example 4.18

whose solution space is one-dimemensional, and is spanned by the vector of coefficients of

the desired implicit equation g. We believe that implicitization is a fruitful direction of

further study in numerical or integer linear algebra.

All our Linux software along with user manual and details about the implementa-

tion can be downloaded from my website, which is currently:

http://math.berkeley.edu/∼jyu

4.4.2 Computing tropical discriminants

Given a d × n matrix A, considered as d points in Zn, the A-discriminant is the

dual variety of the toric variety associated to A. It is an object of great interest in algebraic

geometry and combinatorics [GKZ94]. In [DFS], Dickenstein, Feichtner, and Sturmfels

107

gave a construction for the tropicalization of the A-discriminant. Their main theorem is

the following.

Theorem 4.19. [DFS, Theorem 1.1] The tropical discriminant of A is the Minkowski sum

of T (kernel(A)) and the row space of A.

Hence the key to computing the tropical discriminant is the tropical linear space of

kernel of A. The point configuration A is called non-defective if the (tropical) A-discriminant

is a (tropical) hypersurface. In this case, the defining polynomial of the hypersurface is

called the A-discriminant. We can use the same method as above to construct its Newton

polytope from the tropical discriminant.

Example 4.20. We will compute the mixed discriminant of the two quadrangles in [MO,

Figure 3]. The point configuration is the following

A =

1 1 1 1 0 0 0 0

0 0 0 0 1 1 1 1

1 2 0 0 2 0 1 1

0 0 1 2 1 1 2 0

.

The Bergman fan of kernel(A) has the following rays with a one dimensional

lineality space spanned by (1, 1, 1, 1, 1, 1, 1, 1) and 72 four-dimensional maximal cones. This

is just one possible fan structure of the tropical linear space. The number of rays and cones

will vary depending on the method used to compute them.

RAYS

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

108

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

-1 -1 -1 -1 -1 -1 -1 0

1 1 1 1 0 0 0 0

1 1 0 0 1 1 0 0

1 0 1 0 1 0 1 0

0 -1 0 -1 -1 0 -1 0

0 1 0 1 1 0 1 0

-1 0 -1 0 -1 0 -1 0

-1 -1 0 0 -1 -1 0 0

-1 -1 -1 -1 0 0 0 0

The A discriminant is a polynomial in 8 variables with 222 monomials. Its Newton

polytope has the following 28 vertices and 12 facets.

VERTICES

1 0 3 0 7 2 0 0 6

1 0 6 4 0 0 4 4 0

1 0 4 4 2 4 4 0 0

1 0 2 8 0 4 0 2 2

1 8 0 0 2 2 2 4 0

1 0 6 0 4 2 6 0 0

1 6 0 0 4 5 3 0 0

1 4 0 6 0 5 1 2 0

1 4 2 4 0 0 0 6 2

1 4 0 0 6 4 0 0 4

1 0 2 6 2 5 1 0 2

1 3 0 3 4 5 0 0 3

1 6 0 4 0 2 0 5 1

1 7 0 3 0 2 1 5 0

1 4 4 0 2 0 4 4 0

1 0 4 0 6 0 0 2 6

1 0 7 0 3 0 6 2 0

1 0 4 6 0 0 0 5 3

1 4 2 0 4 0 0 4 4

1 6 0 0 4 2 0 3 3

1 6 2 0 2 0 2 5 1

1 3 4 3 0 0 3 5 0

1 5 2 3 0 0 1 6 1

1 3 0 7 0 5 0 2 1

1 4 0 4 2 6 2 0 0

1 3 0 5 2 6 1 0 1

109

1 0 2 4 4 4 0 0 4

1 0 4 6 0 3 3 2 0

Example 4.21 (Hyperdeterminant of the 3-cube). Let A be the vertices of the 3-cube, i.e.

A =

1 1 1 1 1 1 1 1

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

.

The A-discriminant is

D222 = 4c000c011c101c110 + 4c001c010c100c111

+c2000c

2111 + c2

001c2110 + c2

010c2101 + c2

011c2100

−2c000c001c110c111 − 2c000c010c101c111 − 2c000c011c100c111

−2c001c010c101c110 − 2c001c011c110c100 − 2c010c011c101c100,

which is a polynomial of degree 4 in 8 variables cv where v is a vertex of the 3-cube.

The Bergman fan of the 3-cube contains 80 four-dimensional cones on the following

rays and the lineality space (1, 1, 1, 1, 1, 1, 1, 1).

RAYS

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

-1 -1 -1 -1 -1 -1 -1 0

1 1 1 1 0 0 0 0

1 1 0 0 1 1 0 0

110

0 0 -1 -1 -1 -1 0 0

1 0 1 0 1 0 1 0

0 -1 0 -1 -1 0 -1 0

0 -1 -1 0 0 -1 -1 0

0 1 1 0 0 1 1 0

0 1 0 1 1 0 1 0

-1 0 -1 0 -1 0 -1 0

0 0 1 1 1 1 0 0

-1 -1 0 0 -1 -1 0 0

-1 -1 -1 -1 0 0 0 0

The A-discriminant contains 12 monomials, and the following 6 vertices.

VERTICES

1 0 0 0 2 2 0 0 0

1 0 0 2 0 0 2 0 0

1 2 0 0 0 0 0 0 2

1 0 2 0 0 0 0 2 0

1 0 1 1 0 1 0 0 1

1 1 0 0 1 0 1 1 0

As we increase the dimension, the number of vertices of the Newton polytope is

much less than the number lattice points in it, hence the number monomials in the polyno-

mial. For instance, the A-discriminant of the 4-cube has 2,894,276 monomials, although its

11-dimensional Newton polytope has only 25,448 vertices [HSYY07]. The tropical approach

allows us to compute the Newton polytope of the A-discriminant without first computing

the entire polynomial.

111

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