index [] · the workshop computer algebra and diﬀerential equations (cade 2009) is the...

Index

I Introduction 3

II Programme 7

III Abstracts 13

IV Participants’ addresses 49

1

Introduction

5

Dear participants:

The workshop Computer Algebra and Di!erential Equations (CADE 2009) is thecontinuation of a series of International workshops on computer algebra and itsapplications that started in Dubna (Russia) in 1979, 1982, 1985, 1990 and 2001 andfollowed in Turku (Finland) in 2007. This series of conferences covered all areas ofcomputer algebra, in particular those related to the applications of symbolic andalgebraic computation to ordinary and partial di!erential equations.

The goal of this workshop is to bring together researchers developing and apply-ing computer algebra techniques in the field of di!erential equations. In this way themeeting will be a cross-fertilisation between the di!erent topics of computer algebraand di!erential equations, and will increase communication between the scientistsworking in the theoretical aspects of computer algebra and the researches who usethem, thus encouraging cooperation among the participants. The programme sched-ule will consist of three working days plus an extra day for an excursion to somehistorical sights in Navarra.

The members of the organising committee of the workshop wish that you enjoyyour stay in Navarre.

Programme

9

Wednesday, October 28: Room 04. El Sario Building

9-9:30 Registration

9:30-10:00 Opening session by the President of the State University of Navarre,Dr. Julio Lafuente

10:00-13:30 Morning session, Chairman: Francisco J. Castro-Jimenez

10:00-11:00 A. Bruno and V. Edneral: Integrability analysis of a poly-nomial system of ODEs near a degenerate stationary point

11:00-11:30 A.A. Gusev, O. Chuluunbaatar, V.P. Gerdt, B.L. Markov-ski, V.V. Serov and S.I. Vinitsky: Algorithm for reduction of bound-ary value problems in multistep adiabatic approximation

11:30-12:00 Co!ee break: “Autoridades” Room. El Sario Building

12:00-12:30 A. Abad, R. Barrio, F. Blesa and M. Rodrıguez: A Math-ematica interface for the Taylor series method

12:30-13:00 A. Myllari and S. Slavyanov: Painleve type dynamicalsystem generated by two Coulomb centers problem

13:00-13:30 O. Efimovskaya: Decompositions of the loop algebra overso(4) and integrable models of the chiral equation type

13:30-15:00 Lunch: Restaurant Building

15:30-18:30 Afternoon session, Chairman: Victor Edneral

15:30-16:30 A. Abad and A. Elipe: Evolution strategies for computingperiodic orbits

16:30-17:00 M. Lara and S. Ferrer: A very high order, Mathematicagenerated, Lie series solution to the rigid body motion


17:30-18:00 J.F. Palacian and P. Yanguas: Computer algebra methodsin transition state theory for high rank saddles

18:00-18:30 M. Sansottera: Towards stability results for planetaryproblems with more than three bodies

10

Thursday, October 29: Room 04. El Sario Building

9:30-10:00 Registration

10:00-13:30 Morning session, Chairman: Vladimir Gerdt

10:00-11:00 J.M. Ucha and F.J. Castro-Jimenez: Interesting examplesof quasi free divisors

11:00-11:30 A. Lasaruk and T. Sturm: Approximate reduction of linearpartial di!erential operators


12:00-12:30 N. Vassiliev: Fast polynomial reduction

12:30-13:00 D. Stefanescu, V.P. Gerdt and S. Yevlakov: Numericalcomputation of bounds for positive roots

13:00-13:30 E. Roanes-Lozano, A. Hernando and L.M. Laita: Somereflections inspired by the design of a locomotive classification ex-pert sytem

13:30-15:00 Lunch: Restaurant Building

16:00-18:00 Visit to the Planetarium of Pamplona: projection of the movie“Evolution” (in English) and a walk around the sky

18:00-20:30 Pamplona downtown: we shall visit some remarkable churchesand historical buildings of the city

20:30-22:30 Tapas tasting itinerary around the most emblematic bars of Pam-plona

Friday, October 30: Room 6. El Sario Building

9:30-10:00 Registration

10:00-13:30 Morning session, Chairman: Nikolay Vassiliev

10:00-11:00 V.P. Gerdt and Yu.A. Blinkov: On algorithmic consistencycheck for discretized PDEs

11:00-11:30 A.I. Zobnin: Di!erential polynomials and standard bases

11


12:00-12:30 A. Myllari, T. Myllari, V. Rostovtsev, S. Vinitsky: Formalintegral and caustics in generalised Contopoulos model

12:30-13:00 F. Crespo, G. Dıaz, S. Ferrer and M. Lara: Poisson and sym-plectic normalization of 4-DOF isotropic oscillators. The van der Waalsfamily

13:00-13:30 J.F. San Juan: Analytical model for Lunar orbiter revisited

13:30-15:00 Group picture: El Sario Building entrance. Lunch: Restaurant Build-ing

15:30-18:30 Afternoon session, Chairman: Aleksandr Myllary

15:30-16:30 P. Acosta and J.J. Morales-Ruiz: A Galoisian approach tosupersymmetric quantum mechanics

16:30-17:00 V.P. Gerdt and and Yu.A. Blinkov: Symbolic-numeric studyof finite di!erence schemes for Navier-Stokes equations


17:30-18:00 V.V. Kornyak: Discrete dynamical models with quantum be-haviour

18:00-18:30 Closing session

21:00-23:30 Banquet dinner: Restaurant (cider house) Kaleangora, TajonarStreet, 29, to enjoy one of the most popular gastronomic traditions inNorthern Navarre

Saturday, October 31

9:30-17:30 Day trip along St. James’ way (Camino de Santiago) visiting theRomanesque architecture of Estella, Puente La Reina, Santa Marıa deEunate and the famous vineyards and wine cellars of Senorıo de Sarrıawhere we will have a wine tasting and a delicious meal

Abstracts

15

Poisson and symplectic normalization

of 4-DOF isotropic oscillators.

The van der Waals family

F. Crespo1, G. Dıaz1, S. Ferrer1 and M. Lara2

Abstract

The 4-DOF Van der Waals family of perturbed Hamiltonian oscillators in 1:1:1:1resonance, which includes several models for perturbed Keplerian systems, is studiedwith the help of a computer algebra system. Normalization by Lie-transforms isapproached both in Poisson and symplectic formalisms. The first procedure relieson the quadratic invariants associated to the symmetries, and is based on previouswork of the authors. Hinging on the maximally superintegrable character of theisotropic oscillator, the symplectic reduction is carried out a la Delaunay using ageneralization of those variables to 4-DOF. Explicit expressions of the Delaunaynormalization up to the second order are presented, showing that they may beextended to higher orders.

We then focus on the search of the relative equilibria and the bifurcations withboth treatments. Due to the symmetries of the system there are isolated as well ascircles of stationary points. This feature manifests rather di!erently in the analysisof the normalized flow in the formulations considered, due to the constraints amongthe invariants versus the singularities associated to the Delaunay chart. The prosand cons of each approach are presented in detail.

[1] Cushman, R.H., and Bates, L.M., Global Aspects of Classical Integrable Sys-tems, Birkhauser Verlag, Basel, 1997.

[2] Egea, J., Sistemas Hamiltonianos en Resonancia 1:1:1:1. Reducciones Toroi-dales y Bifurcaciones de Hopf. Tesis Doctoral. Universidad de Murcia, 2007.

[3] Egea, J., Ferrer, S., and Van der Meer, J.C., Hamiltonian fourfold 1:1 res-onance with two rotational symmetries, Regular and Chaotic Dynamics 12,664-674, 2007.

1Dpto de Matematica Aplicada, Universidad de Murcia, Spain,[email protected], [email protected], [email protected], http://www.um.es/

2Real Observatorio de la Armada, San Fernando, Spain, [email protected]

16

[4] Dıaz, G., Egea, J., Ferrer, S., Van der Meer, J.C., and Vera, J.A., RelativeEquilibria and Bifurcations in the generalized Van der Waals 4-D Oscillator,submitted to Physica D, 2009.

[5] Crespo, F., Osciladores isotropos 4-D perturbados. Normalizaciones Poissony Simplecticas. Aplicaciones. Tesis doctoral en preparacion. Universidad deMurcia, 2009.

17

Integrability analysis of a

polynomial ODE system near a

degenerate stationary point

A.D. Bruno1 and V.F. Edneral2

We study integrability of the autonomous system

dx/dt = !y3 ! b x3y + a0 x5 + a1 x2y2,

dy/dt = (1/b) x2y2 + x5 + b0 x4y + b1 x y3,(1)

i. e. we look for conditions on parameters a0, a1, b, b0, b1 which guarantee integra-bility. In the case when the first quasi-homogeneous polynomial approximation isHamiltonian, with the additional assumption that the polynomial Hamiltonian isexpandable into the product of only square-free factors, the problem is solved in [1].Therefore here we discuss only non Hamiltonian case of (1).

Systems with a nilpotent matrix of the linear part were thoroughly studied byLyapunov et. al. In the example above, there is no linear part, and the firstapproximation is not homogeneous but quasi homogeneous. This is the simplest caseof a planar system without linear part where the Newton’s open polygon consists ofa single edge. In general case, such problem was not studied.

To study local integrability of the system near a degenerate stationary pointx = 0, y = 0, we use an approach based on the Power Geometry method [2] and onthe computation of the resonant normal form [3,4,5]. We exclude from the analysisthe value b2 = 2/3.

For the planar 5-parametric system (1), we found the complete set of necessaryconditions on parameters of the system for which the system is locally integrable neara degenerate stationary point. It consists of 4 two-parametric sets in {a0, a1, b, b0, b1}space. For 3 such sets we found by independent methods su"cient conditions of alocal integrability. Because these methods are constructive we get for these 3 setsfirst integrals of the system (1). For the forth set we have at the moment onlyapproximations of the local integrals as truncated power series in parameters of thesystem, but we believe that it is possible to sum them up to finite functions.

1Keldysh Institute for Applied Mathematics of RAS, Miusskaya Sq. 4, Moscow, 125047, Russia,[email protected]

2Skobeltsyn Institute of Nuclear Physics of Lomonosov Moscow State University, LeninskieGory 1, Moscow, 119991, Russia.The grant of the Russian Foundation for Basic Research 08-01-00082 and the grant of the Presidentof the Russian Federation for support of scientific schools 195.2008.2.

18

[1] Algaba, A., Gamero, E., and Garcıa, C., The integrability problem for a classof planar systems, Nonlinearity 22, 395–420, 2009.

[2] Bruno, A.D., Power Geometry in Algebraic and Di!erential Equations, Fiz-matlit, Moscow, 1998 (in Russian), Elsevier Science, Amsterdam, 2000 (inEnglish).

[3] Bruno, A.D., Local Methods in Nonlinear Di!erential Equations, Nauka, Mos-cow, 1979 (in Russian), Springer-Verlag, Berlin, 1989 (in English)

[4] Bruno, A.D., and Edneral, V.F., Algorithmic analysis of local integrability,Dokl. Akademii Nauk 424 (3), 299-303, 2009 (Russian), Doklady Mathem. 79(1), 48-52, 2009 (English).

[5] Edneral, V.F., On algorithm of the normal form building. In: Proc. CASC2007. Ganzha et al. (Eds.), Springer-Verlag series, Lecture Notes in ComputerScience (LNCS) 4770, 134–142, 2007.

19

Evolution strategies for computing

periodic orbits

A. Abad1 and A. Elipe1

Abstract

An evolution strategy algorithm belonging to the general field of genetic algo-rithms is developed to detect periodic orbits in dynamical problems. The algorithmis applied to the problem of motion of a particle under the gravitational field of asolid circular wire.

1Grupo de Mecanica Espacial and IUMA, Universidad de Zaragoza, Spain,{abad,elipe}@unizar.es

20

On algorithmic consistency check for

discretized PDEs

V.P. Gerdt1 and Yu.A. Blinkov2

In the talk we use a general algorithmic approach [1] to discretization of a systemof PDEs which admit the conservation law form. It is assumed that the initial PDEsystem is involutive. Otherwise, before doing discretization the system has to becompleted to involution. An important example of such a system is the Navier-Stockes equations whose discretization considered recently in [2].

We argue that a di!erence Grobner basis, when it can be constructed in thecourse of the finite-di!erence scheme generation [1,2], contains valuable informationon the scheme consistency. Thus, in the case of consistent discretization, a subset ofthe basis containing no partial derivatives yields in the continuous limit a di!erentialequation which can be obtained from the initial involutive di!erential system bydoing prolongations and projections (reductions). Some examples of generationof consistent and inconsistent schemes are considered, and the related algorithmicaspects of the consistency check are analyzed.

[1] Gerdt, V.P., Blinkov, Yu.A., and Mozzhilkin, V.V., Grobner bases and gener-ation of di!erence schemes for partial di!erential equations. Symmetry, Inte-grability and Geometry: Methods and Applications (SIGMA) 2, 051, 26 pages,(2006), arXiv:math.RA/0605334

[2] Gerdt, V.P., and Blinkov, Yu.A., Involution and di!erence schemes for theNavier-Stokes equations, Lecture Notes in Computer Science (LNCS) 5743,94-105, 2009, Springer-Verlag, Berlin.

1Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia,[email protected], http://www.jinr.ru

2Department of Mathematics and Mechanics, Saratov State University, Saratov, Russia,[email protected], http://www.sgu.ru

21

Symbolic-numeric study of finite

di!erence schemes for Navier-Stokes

equations

V.P. Gerdt1 and Yu.A. Blinkov2

In the talk we present the results of numerical reconstruction of an exact solutionto the two-dimensional incompressible Navier-Stokes equations given in [1]. The re-construction is based on two finite-di!erence schemes obtained fully algorithmicallyby applying the Grobner bases method to the straightforward discrete version of theoriginal di!erential equations as described in [2]. The specific feature of both di!er-ence schemes is automatic fulfillment of the discrete version of continuity equationon the next temporal layer.

[1] Kim J., and Moin P., Application of a fractional-step method to incompressibleNavier-Stokes equations, J. Comp. Phys. 59, 308-323, 1985.

[2] Gerdt, V.P., and Blinkov, Yu.A., Involution and di!erence schemes for theNavier-Stokes equations. LNCS 5743, 94-105, 2009, Springer-Verlag, Berlin.

1Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia,[email protected], http://www.jinr.ru

2Department of Mathematics and Mechanics, Saratov State University, Saratov, Russia,[email protected], http://www.sgu.ru

22

Algorithm for reduction of

boundary-value problems in multistep

adiabatic approximation

A.A. Gusev1, O. Chuluunbaatar1, V.P. Gerdt1, B.L. Markovski1, V.V. Serov2 andS.I. Vinitsky1

Abstract

The adiabatic approximation is well-known method for e!ective study of few-body systems in molecular, atomic and nuclear physics. On the base of pioneeringwork of Born and Oppenheimer (1927) the method was applied in various problemsof physics, using the idea of separation of ”fast” and ”slow” variables.

Purpose of this talk is to present algorithm for generalization of the standard adi-abatic ansatz for the case of multi-channel wave function when all variables treateddynamically [1,2]. For this reason we are introducing the step-by-step averagingmethods for order to eliminate consequently from faster to slower variables.

We present a symbolic algorithm for reduction of multistep adiabatic equations,corresponding to the MultiStep Generalization Kantorovich Method (MSGKM), forsolving multidimensional boundary-value problems [3].

Algorithm MSGKM

Input:H =

!Ni=1 HN+1!i is initial Hamiltonian dependent on ordered variables !x = {xN "

xN!1 " ... " x1}T decomposed to sum of partial Hamiltonians Hi # Hi(xi; xi!1, ...,x1), dependent on subset “faster” xi and “slower” xi!1, ..., x1 variables;H"n1 ! En1"n1 = 0, $n"

1|n1% ="

dxN ...dx1"†n!1

(!x)"n1(!x) = #n!1n1

is main eigenvalue problem for calculation of $!x|n1% # "n1(!x) and En1 = $n1 .Output:A set Eq(k) k = 1, ..., N is a set of auxiliary parametric eigenvalue problems forcalculation of "nk

# "(k)nk

(xN , ..., xk; xk!1, ..., x1) and $nk# $(k)

nk# $(k)

nk(xk!1...x1),

where # = "(1)n1

and En1 = $(1)n1

are solutions of the main eigenvalue problem.Local:"(k)

nk# "(k)

nk(xN , ..., xk; xk!1, ..., x1) and $nk

# $(k)nk# $(k)

nk(xk!1...x1) are solutions of

the auxiliary parametric eigenvalue problems(!N

i=N+1!k HN+1!i)"(k)nk! $(k)

nk"nk

= 0,

1Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia, [email protected] State University, Saratov, 4100012, Russia

23

$n"k|nk% #

"dxN ...dxN+1!k"

†n!k

"nk

$n"k+1|nk% # %nk+1nk

(xk; xk!1, ..., x1) are auxiliary solutions:$n"

k+1|nk% ="

dxN ...dxk+1"(k+1)nk+1

"(k)nk

.

1: Eq(N):={HnN |nN% ! $nN |nN% = 0, $"(N)†nN

|"(N)n!N% = #nNn!N

}2: Eq(N) &{|nN%, $nN}3: for k:=N-1:1 step -14: Eq(k):={($(k+1)

nk+1! $(k)

nk+ Hk)$nk+1|nk%

+!

n!k+1$nk+1|

#Hk, n"

k+1%$$n"

k+1|nk%=0}.5: Eq(k) &{$nk+1|nk%, $(k)

nk}

6: |nk% :=!

n!k+1|n"

k+1%$n"k+1|nk%

7: end for8: # = |n1%, En1 = $(1)

n1

The work was supported by RFBR (grants 07-01-00660, 08-01-00604).

[1] Makarewicz, J., Adiabatic multi-step separation method and its applicationto coupled oscillators, Theor. Chim. Acta 68, 321-334, 1985.

[2] Dubovik, V.M., Markovski, B.L., and Vinitsky, S.I., Multistep adiabatic ap-proximation, Preprint JINR E4-87-743, (Dubna, 1987),http://www-lib.kek.jp/cgi-bin/img_index?8801189.

[3] Chuluunbaatar, O., Gusev, A.A., Abrashkevich, A.G., Amaya-Tapia, A., Kas-chiev, M.S., Larsen, S.Y., and Vinitsky, S.I., KANTBP: A program for com-puting energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach, Comput. Phys. Commun. 177,649-675, 2007.

24

Some reflections inspired by the design of

a locomotive classification expert system

E. Roanes-Lozano1, A. Hernando2 and L.M. Laita3

Abstract

There are many railway fans around the world who love watching and takingphotographs of trains (there is even a curious hobby involving collecting sightingsof trains, usually of a certain set of moving stock: “trainspotting). Such fans wouldconsider useful to have an automated locomotive identification system, especiallywhen traveling abroad. Thinking of the design of such a system sparked somereflections on the standard Rule Based Expert System (RBES) design. Clearly, theapproaches detailed here have other applications beyond these hobbies, like: targetrecognition; aerial photograph analysis; fossil, animal and plant classification; etc.

We have here chosen to focus on an interesting period of the Spanish broadgauge railways: the 70s, where all main line diesel locomotives (14 classes) werepainted either in Renfe (Spanish broad gauge railway company) livery: green withtwo horizontal yellow thin stripes (11 classes) or Talgo livery: metallic silver witha thick red horizontal stripe (3 classes). The shunting diesel locomotives of thisperiod are also painted in Renfe livery, but do not have bogies, so they can beeasily recognized from the main line ones, that are all of the BB or CC types.We have focused on this period because the sample is complex enough for easilyperforming some testing and for showing the possibility to build such a RBES,while relevant enough to make the classification easy. In fact, we have consideredmore than a dozen features for the locomotives. We have already used in earlierworks an algebraic inference engine for RBES knowledge extraction and knowledgeverification (based on the use of Grobner bases). Coronary artery bypass surgerytechniques; anorexia, depression and fybromialgia diagnosis, ... are topics alreadytreated. We have also used this approach to RBES design and development here,but from two di!erent points of view. Both implementations are written in Maple.

(1) The first one is a classic rule based expert system that is trivially built froma flowchart designed by experts. A drawback is that it depends too far on

1Depto. Algebra, Universidad Complutense de Madrid, Spain, [email protected],http://www.ucm.es/info/secdealg/ERL

2Depto. Sistemas Inteligentes Aplicados, Universidad Politecnica de Madrid, Spain,[email protected]

3Depto. Inteligencia Artificial, Universidad Politecnica de Madrid, Spain, [email protected]

Partially supported by the research projects TIN2009-07901 (Ministerio de Educacion y Ciencia,Spain) and UCM2008-910563 (UCM - BSCH Gr. 58/08, research group ACEIA).

25

the experts knowledge, as they are in charge for the laborious and not trivialtask of manually selecting the shape of the flowchart and the features whichmust be asked. Indeed, the experts impose their criteria to the design of theflowchart, whereas the classification could normally be based on other features(that could be precisely the ones visible, for instance, at a certain photograph).However, this approach has the advantage of running very fast.

(2) The second one di!ers from the usual RBES design. Each locomotive classimplies a conjunction of features. At any moment, when we have alreadyobserved some set of features, we can ask if it already determines any specificmodel or there are still more than one model satisfying those conditions. Theadvantage of this approach is the freedom to perform a process analogous toforward firing with incomplete data (and without using a three-valued logic).

26

Discrete dynamical models with

quantum behavior

V.V. Kornyak1

Abstract

We construct a class of discrete dynamical models allowing quantum description.The models are defined on regular graphs with transitive groups of symmetries. Thevertices of the graphs take values in finite sets of local states. The evolution of themodels proceeds in the discrete time. We assume that one-time-step quantum transi-tions are allowed only within the neighborhoods of the graph vertices. Our approachto quantization consists in introduction of gauge connection with values in unitaryrepresentation of finite group: the elements of the connection are interpreted asamplitudes of quantum transitions. In this approach all our manipulations — incontrast to the standard quantization — remain within the framework of construc-tive discrete mathematics requiring no more than the ring of algebraic integers. Insome cases the models under consideration can be approximated by partial di!er-ential equations. Essential part of our study was carried out with the help of aprogram in C — implementing computer algebra and computational group theoryalgorithms — we are developing now.

1Laboratory of Innformation Technologies, Joint Institute for Nuclear Research, Dubna, Russia,[email protected], http://compalg.jinr.ru/CAGroup/Kornyak/

27

A very high order, Mathematica c'

generated, Lie series solution to the

rigid body motion

S. Ferrer1 and M. Lara2

Abstract

As is well known since the times of Jacobi, and is discussed in most classicalmechanics textbooks, the di!erential equations of the Euler-Poinsot problem can besolved analytically. However, the integration of rigid body motion still continues tobe a fundamental problem in rotational dynamics for a variety of reasons. Amongthem, we can mention the complexity of carrying elliptic functions and integrals overa perturbation scheme, and the lack of e"ciency one may found in the numericalevaluation of these special functions [1]. Then, the numerical integration of torque-free rotation may appear as a competitive alternative to the exact analytical solution[2].

The solution in elliptic integrals is essential to the rigid body motion. Never-theless, the fact that the Hamiltonian of the Euler-Poinsot problem can be splitinto two integrable parts —a term that describes the integrable motion of an ax-isymmetric rigid body, and an integrable part describing the triaxial deviation fromaxisymmetry— may allow to completely avoid the use of elliptic functions [3]. In-deed, for those cases in which the triaxiality of the rigid body is small, we can applya perturbative approach to obtain an analytical solution that is a compact series intrigonometric functions of straightforward evaluation.

The convergence of the series solution may be poor when the departure from ax-isymmetry is not so small, thus requiring very high orders to make the perturbationsolution competitive. However, we reach these higher orders with Deprit’s pertur-bation method [4], which is specifically designed for the automatic computation ofthe Lie series solution. As expected, the lower order terms of our series match thosein the literature obtained by the Taylor series expansion of the closed form solution.But our perturbation approach is clearly simpler because it completely does withoutelliptic functions.

We carry out all the computations with Mathematica c" and get a formal seriessolution that is complete up to the 25th order in the small parameter (the triaxiality

1Dpto de Matematica Aplicada, Universidad de Murcia, Spain, [email protected],http://www.um.es

2Real Observatorio de la Armada, San Fernando, Spain, [email protected]

28

coe"cient). In addition, we left the algebraic manipulator the responsibility ofsolving by itself the quadratures and partial di!erential equations that appear inthe procedure, a duty that not so long ago had to be assumed by the user, whichhad to implement by hand the required replacing rules.

Comparison of our series solution with the exact solution in the case of di!erentSolar system bodies, shows that the numerical errors can be confined to the ma-chine’s arithmetic precision even with relatively low orders. Thus, a fourth ordersolution is required for the Earth, while a tenth order is needed for Mars. On thecontrary, di!erent trials using the inertia parameters of the the Moon show that atwenty-first order truncation may be required for reaching the computer numericalprecision.

[1] Fukushima, T., Fast computation of Jacobian elliptic functions and incompleteelliptic integrals for constant values of elliptic parameter and elliptic charac-teristic, Celestial Mechanics and Dynamical Astronomy, 2008, DOI 10.1007/s-008-9177-y.

[2] Fukushima, T., Simple, regular, and e"cient numerical integration of rota-cional motion, The Astronomical Journal, 135, 2298-2322, 2008.

[3] Touma, J., and Wisdom, J., Lie-Poisson integrators for rigid body dynamicsin the Solar System, The Astronomical Journal, 107, 1189–1202, 1994.

[4] Deprit, A., Canonical transformations depending on a small parameter, Ce-lestial Mechanics, 1, 12-30, 1969.

29

A Galoisian approach to Supersymmetric

Quantum Mechanics

J.J. Morales-Ruiz1

Abstract

(Joint work with Primitivo B. Acosta-Humanez)

In this work we will show that the Galois Theory of Linear Di!erential Equations- also called the Picard-Vessiot Theory - is the natural framework to study thesolvability in closed form of the Schrodinger equation of Quantum Mechanics. Thus,all the cases where this equation was solved by quadratures can be classified usingthe Picard-Vessiot Theory. As a by product, we obtain a Galoisian interpretationof the Supersymmetric Quantum Mechanics.

1Departamento de Matematica e Informatica Aplicadas a la Ingenierıa Civil, E.T.S.I. deCaminos, Canales y Puertos, Polytechnic University of Madrid, Spain, [email protected]

30

Painleve type dynamical system

generated by two Coulomb centers

problem

A. Myllari1 and S. Slavyanov2

Abstract

We study a specific dynamical problem generated by the process of “antiquanti-zation” for the quantum two Coulomb centers problem. Since in the latter problemstationary states can be considered, no physical time is presented in our dynami-cal problem; the distance between centres is regarded as the independent variable.Therefore, the motion is considered as variation of coordinates and momentumsalong with variation of this distance. It is important that the corresponding hamil-tonian system is a"tliated to the Painleve-type equations, namely to a particularcase of P 5. That means that this system is integrable and also that we can speakabout existence of explicit relationship between behavior of solutions at di!erent sin-gular points of the system. To a certain extent it is equivalent to explicit knowledgeof scattering characteristics.

1Department of Physics and Astronomy, University of Turku, Finland, [email protected] of Physics, St. Petersburg State University, Russia

31

Formal integral and caustics in

generalised Contopoulos model

A. Myllari1, T. Myllari1, A. Rostovtsev2 and S. Vinitsky2

Abstract

We continue our studies of the feasibility of the usage of formal integral in in-vestigating structure of caustics in dynamical systems with two degrees of freedom.Generalised Contopoulos model is used for experiments. We use the program LINA(version for Mathematica 6) developed at JINR to construct Gustavson-like formalintegral of motion. This integral is used (together with the Hamiltonian of the sys-tem) to study analytically the evolution of caustics (changes in the structure of thevelocity field) in the system. Results obtained analytically by using formal integralof motion are compared with the ones obtained by the numerical integration.

1Department of Physics and Astronomy, University of Turku, Finland, [email protected] Institute for Nuclear Research, Dubna, Russia

32

Computer algebra methods in

transition state theory for

high rank saddles

J.F. Palacian1 and P. Yanguas1

Abstract

(Joint work with Charles Ja!e, George Haller and Turgay Uzer)

In the study of transport problems in di!erent fields such as chemical reactions,atomic physics or celestial mechanics, it is a well known fact that a important pieceof the dynamics is characerised by the existence of equilibrium points of saddle type.Related to these unstable points there are geometric structures such as separatricesand transition states that regulate the transport in phase space. The case of rank-1saddles has been discussed [5]. In this presentation we describe the relevant dynamicsnearby rank-n saddle points where n > 1. The crucial point is to introduce anon-compact hyperbolic invariant manifold using the methodology based on pseudomanifolds [1,2]. We present a proof based on [4] about the actual existence of thishyperbolic manifold, which is the analogous of the normally hyperbolic invariantmanifold that appears in the case n = 1.

In order to compute this hyperbolic manifold and other related structures weneed to use a combination of analytical and numerical tools based on normal forms.In this context, we will describe the computational aspects that allows one to buildthe geometric structures with high accuracy. An important issue is also the visual-isation of the structures when the dimension of the problem is high. For instance,in the problem of helium ionisation process that we will use as our prototype, theHamiltonian of the problem defines a system of five degrees of freedom and thehyperbolic invariant manifold has dimension eight. Thus, we will show how to dealwith this high-dimensional manifolds using di!erent strategies based on suitableprojections.

Finally, we will describe briefly the extension of the above theory to periodicallytime-dependent systems, where external forces to the system (e.g., electric fieldsor the perturbation caused when the Sun is added to a restricted problem, etc.)play an important role [3]. In this situation one needs to use Floquet theory to deal

1Dpt. Ingenierıa Matematica e Informatica, Universidad Publica de Navarra, Spain,{palacian,yanguas}@unavarra.es, http://www.unavarra.es

33

with time-dependent Hamiltonians. Usually, these normal forms are computed usingTaylor-Fourier series and the number of terms increase exponentially from order toorder. Thus, a judicious algorithm to deal with these expressions is introduced,which allows one to handle normal forms even when the number of Fourier terms ishigh.

[1] Haller, G., Uzer, T., Palacian, J.F., Yanguas, P., and Ja!e, C., Transitionstates near rank-two saddles: correlated electron dynamics of helium, Comm.Nonlinear Sci. Num. Simul. 15, 48-59, 2010.

[2] Haller, G., Uzer, T., Palacian, J.F., Yanguas, P., and Ja!e, C., Transitionstate geometry near higher-rank saddles in phase space, submitted, 2009.

[3] Haller, G., Uzer, T., Palacian, J.F., Yanguas, P., and Ja!e, C., Time-dependenttransition state geometry near higher-rank saddles, in preparation.

[4] Irwin, M.C., A new proof of the pseudostable manifold theorem, J. LondonMath. Society 21, 557-566, 1980.

[5] Uzer, T., Ja!e, C., Palacian, J.F., Yanguas, P., and S. Wiggins, The geometryof reaction dynamics, Nonlinearity 15, 957-992, 2002.

34

A Mathematica interface for theTaylor series method

A. Abad1, R. Barrio1, F. Blesa1 and M. Rodrıguez1

Abstract

Our group is developing a new free software called TIDES [4] that is based onthe classical Taylor series method [1,2] using an optimized variable-stepsize variable-order formulation. This software consists on a library on C and a precompiler donein Mathematica that creates a C program (that can also be called by FORTRAN)that permits to compute up to any precision level (by using multiple precision li-braries for high precision when needed) the solution of an ODE system. The softwarehas been done to be extremely easy to use. The program also permits to computein a direct way not only the solution of the di!erential system, but also the partialderivatives, up to any order, of the solution with respect to the initial conditions orany parameter of the system [3].

In this talk we present the symbolic methods used to write, automatically, theC or FORTRAN code that integrates a particular ODE by means of the Taylormethod. These methods are based on the automatic di!erentiation processes andthey are implemented in the Mathematica package MathTIDES.

[1] Barrio, R., Performance of the Taylor series method for ODEs/DAEs, Appl.Math. Comput. 163, 525-545, 2005.

[2] Barrio, R., Blesa, F., and Lara, M., VSVO formulation of the Taylor methodfor the numerical solution of ODEs, Comput. Math. Appl. 50, 93–111, 2005.

[3] Barrio, R., Sensitivity analysis of ODE’s/DAE’s using the Taylor series method,SIAM J. Sci. Comput. 27, 1929-1947, 2006.

[4] Abad, A., Barrio, R., Blesa, F., and Rodrıguez, M., TIDES: a Taylor seriesIntegrator for Di!erential EquationS, Preprint (2009).

1Grupo de Mecanica Espacial, Universidad de Zaragoza, Spain,{abad,rbarrio,fblesa,marcos}@unizar.es, http://gme.unizar.es/software/tides

35

Analytical model for Lunar orbiter

revisited

J.F. San Juan1

Analytical theories based on Lie–Deprit transforms are being used so as to sim-plify the search for families of periodic orbits around planets, natural satellites orasteroids. Normalized equations of motion permit locating the frozen orbit familiesdepending on values of the inclination, eccentricity and semimajor axis. From an-alytical theories, it is possible to derive a robust and easy-to-use tool for missionplanning, whenever the appropiate use of the force model captures the qualitativebehaviour of the problem.

In the case of an orbiter around the Moon, predictions based on recent gravitypotential and short analytical theories can produce significant di!erences in long-term behaviour to others obtained with previous gravity potential models [3,5].

In this sense, this work revisits the analytical theory presented in [1] which wasdeveloped to analyze long-term behaviour of low near-circular orbits. This analyticaltheory only took into account the J2 and J7 zonal contribution, even when theconclusions obtained are valid for some of the available lunar gravity models [2],they are not exactly valid for more recent models (LP150Q). In particular, thistheory does not depict the families of frozen orbits for inclinations greater than 69#

as shown in Figure 1. This is because in the case of the Moon the absolute valuesof the harmonic coe"cients do not decrease when increasing the degree and order.

In this paper we analyze the motion of an orbiter around the Moon perturbedfrom J2 up to J20 zonal and C2,2 tesseral harmonics coe"cients [4]. By means ofthree Lie transforms we produce a closed-form fourth-order analytical theory. Thenew Hamiltonian only depends on the argument of the perilune and its long-termbehaviour can then be studied. The contribution of the zonal harmonics appears inthe second order whereas the contribution of C2,2 appears in the fourth order. Wealso want to note that the C2,2-terms do not a!ect long-term behaviour. Finally,the influence of the short and medium-terms can be recovered from the inversetransformations.

1Departamento de Matematica y Computacion, Universidad de La Rioja, Spain, [email protected], http://www.unirioja.es

36

Figure 1: Moon frozen orbits surface as a function of the eccentricity, semimajor axis andinclination for J2 and J7 zonal contribution (left) and C2,2 tesseral and from J2 up to J20

zonal contribution (right). The color of the surface represents the perigee argument value.Red represents g = !/2 and green g = 3!/2.

[1] Abad, A., Elipe, A., San Juan, J. F., and San-Martın, M., An analytical model forLunar orbiter, Acta Academiae Aboensis, Ser. B 67, 134-143, 2007.

[2] Elipe, A., and Lara, M., Frozen orbits about the Moon, Journal of Guidance, Con-trol and Dynamics 26, 238–243, 2003.

[3] Floberghagen, R., Lunar Gravimetry, Astrophysics and Space Science Library 273,Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.

[4] De Saedeleer, B., Analytical theory of a Lunar artificial satellite with third bodyperturbations, Celest. Mech. Dynam. Astron. 95, 407-423, 2006.

[5] Lara, M., and Ferrer, S., Lunar analytical theory for polar orbits in a 50-degreezonal model, AAS/AIAA Space Flight Mechanics Meeting. Savannah, Georgia,February 9–12, Paper AAS 08-184, 2009.

37

Towards stability results for planetary

problems with more than three bodies

M. Sansottera1

Abstract

(I carried out this research work jointly with A. Giorgilli and U. Locatelli)

We study the stability of the planetary problem including Sun, Jupiter, Saturnand Uranus (SJSU, respectively). We adopt a similar approach to that success-fully used (in the past few years) to construct invariant tori for the SJS three bodyproblem in a semi–analitic way. We first make classical expansions of all the in-teraction terms between each pair of planets appearing in the Hamiltonian of theSJSU system. Therefore, we remove the main perturbing terms depending on themean motion angles by a couple of canonical transformations similar to those usu-ally adopted in the Kolmogorov’s scheme for the construction of KAM tori. Allthese expansions are explicitly performed by algeraic manipulation on a computer.In order to reduce the huge number of terms to handle in the expansions, we limitourselves to study the SJSU problem in the planar case.

Since we want to investigate the results we can produce in the framework of thenormal form theory, as a first stressing test, we furtherly simplify our model in adrastic way, by averaging the Hamiltonian with respect to the mean motion angles.Thus, we obtain a three degrees of freedom model, which is able to approximatethe secular motions up to order 2 in the masses (and to high degree, i.e. 12, in theeccentricities). By applying the standard estimates to the partial construction of theBirkho! normal form, we show that our secular Hamiltonian model is “e!ectivelystable” for times larger than the age of the solar system for all the initial conditionscorresponding to eccentricities smaller than (about) 1/2 of the real values.

1Dipartimento de Matematica, Universita degli Studi di Milano, Italy,[email protected], http://newrobin.mat.unimi.it/users/sansotte/

38

Numerical computation of bounds

for positive roots

Doru Stefanescu1, V.P. Gerdt2 and S. Yevlakov2

Abstract

We discuss the numerical computation of positive roots for univariate polyno-mials with real coe"cients. The estimation of the bounds for positive roots ofunivariate polynomials with real coe"cients is a main step for the computation ofreal roots.

We shall give a general bound for positive roots of univariate polynomials withreal coe"cients. The coe"cents of such polynomials have at least a sign variation.So they can be represented as in the next result.

Theorem. Let

P (X) = a1Xd1 + a2X

d2 + · · · + asXds ! b1X

e1 ! b2Xe2 ! · · ·! btX

et ( R[X] ,

where ai > 0 , bj > 0, d1 = deg(P ) and d1 > d2 > · · · > ds .An upper bound for the positive roots of P is given by

max1#i#s1#j#t!j $=0

%&jibj

'jai

& 1

di ! ej

for any 'j ) 0 , &jk ) 0 such that

!tj=1 'j * 1 ,

!si=1 &ji ) 1 with &ji = 0 if di < ej .

Then we derive the numerical computational complexity of our estimatations ofthe bounds and make experimental comparisons of our bounds with other knownbounds from the literature.

The knowledge of accurate bounds for real roots is significant for the computa-tion of real roots, because they allow to have good approximations of the intervalcontaining real roots. In particular the estimation of lower bounds for positive roots

1University of Bucharest, Romania, [email protected], http://rms.unibuc.ro/stef2Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia,

{gerdt, evlakhov}@jinr.ru, http://compalg.jinr.ru/CAGroup/Gerdt/

39

is a key step in real root isolation. Since a lower bound for the postive roots is theinverse of an upper bound for the positive roots it is su"cient to estimate the upperbounds.

As benchmarks we choose some classical orthogonal polynomials and families oflacunary polynomials.

[1] Abramov, S.A., Lectures on Complexity of Algorithms (in Russian), MCCME2008.

[2] Akritas, A., Linear and quadratic complexity bounds on the values of thepositive roots of polynomials, Univ. J. Comput. Sci. 15, 523-537, 2009.

[3] Kioustelidis, J.B., Bounds for positive roots of polynomials, J. Comput. Appl.Math. 16, 241-244 1986.

[4] M. Mignotte, and D. Stefanescu, Polynomials – An Algorithmic Approach,Springer Verlag, 1999.

[5] D. Stefanescu, New bounds for positive roots of polynomials, Univ. J. Comput.Sci. 11, 2125-2131, 2005.

[6] D. Stefanescu, Computation of dominant real roots of polynomials, Prog. andCompt. Sotfware 34, 69-74, 2008.

[7] Sharma, V., Complexity of real root isolation sing continued fractions, Theor.Comp. Sci. 409, 292-310, 2008.

[8] Tsigaridas, E., and Emiris, I., On the complexity of real root isolation usingcontinued fractions, Theor. Comp. Sci. 392, 158-173, 2008.

40

Approximate reduction of linear

partial di!erential operators

A. Lasaruk1 and T. Sturm2

Abstract

We apply real quantifier elimination procedures to find necessary and su"cientconditions for the existence of certain factorizations of linear partial di!erentialoperators over the field of multivariate rational functions together with ordinaryformal partial derivations. In the positive case of reducibility extended real quantifierelimination yields at least one sample factorization.

We generally restrict to the special case of detecting left factors of order one.Furthermore, our factorizations restrict the syntactical form of the coe"cients ofthe admissible factors in some way definable by first-order formulas in the Tarskialgebra, i.e. the theory of the reals with arithmetic and ordering.

In addition to exact factorization in the sense above we consider approximatefactorization the following sense: The non-commutative product of the obtainedfactors need not equal the original operator but only be close to it with respect tosome suitable norm definable in the Tarski algebra. Furthermore, this property needonly hold over some user-specified semialgebraic region.

We reinterpret results of Grigoriev and Schwarz [2] and Beals and Kartashova [1]to express exact and approximate factorization conditions of a given operator as first-order formulas within the Tarski algebra. Our factorization framework considerablyextends existing work in that area [1,3,4]: Our methods are in theory generallyapplicable to operators of arbitrary order and number of variables. We derive notonly factorization conditions but also sample factorizations. We admit arbitrarysemi-algebraic sets for restricting the region over which factorization takes place.

We have realized a package for computing with linear partial di!erential oper-ators in the open source3 computer algebra system Reduce. This package includesour factorization methods based on Redlog, which is the first-order logic componentof Reduce. We illustrate our work by concrete example factorizations using thispackage.

[1] Beals, R., and Kartashova, E., Constructive factorization of LPDO in twovariables. Journal of Theoretical and Mathematical Physics 145, 1510-1523,2005.

1FORWISS, Universitat Passau, 94030 Passau, Germany, [email protected] de Matematicas, Estadıstica y Computacion, Universidad de Cantabria, 39071

Santander, Spain, [email protected], http://personales.unican.es/sturmt/3http://reduce-algebra.sourceforge.net/

41

[2] Grigoriev, D., and Schwarz, F., Factoring and solving linear partial di!erentialequations, Computing 73, 179-197, 2004.

[3] Kartashova, E., and McCallum, S., Quantifier elimination for approximatefactorization of linear partial di!erential operators. In: Proceedings of theCalculemus 2007. Manuel Kauers (Ed.), Springer-Verlag series: Lecture Notesin Artificial Intelligence (LNAI) 4573 106-115, 2007.

[4] Shemyakova, E., and Winkler, F., A full system of invariants for third-orderlinear partial di!erential operators in general form. In: Computer Algebra inScientific Computing. Proceedings of the CASC 2007. V.G. Ganzha, E.W.Mayr, and E.V. Vorozhtsov (Eds.), Springer-Verlag series: Lecture Notes inComputer Science (LNCS) 4770 360-369, 2007.

42

Interesting examples of quasi free divisors

J.M. Ucha1 and F.J. Castro-Jimenez1

Abstract

Let us denote X = Cn and O = OX the sheaf of holomorphic functions on X.Let D + X be a divisor (i.e. a hypersurface) and p ( X. Denote by Der(Op)the Op-module of C-derivations of the ring of germs Op. According to [K. Saito,Theory of logarithmic di!erential forms and logarithmic vector fields. J. Fac Sci.Univ. Tokyo 27:256-291, 1980] a vector field # ( Der(Op) is said to be logarithmicwith respect to D if #(f) = af for some a ( Op, where f is a local equation of thegerm (D, p) + (X, p). The Op-module of logarithmic vector fields (or logarithmicderivations) is denoted by Der(! log D)p and it is a Lie algebra under the bracketproduct [!,!].

Definition 0.0.1 The divisor D is said to be free at the point p ( D if the Op-module Der(! log D)p is free. The divisor D is called free if it is free at each pointp ( D.

For each divisor D + X we have an inclusion iD : $•(log D) & $•((D) where$•((D) is the meromorphic de Rham complex and $•(log D) is the de Rham log-arithmic complex, both with respect to D. A meromorphic form ) ( $p((D)is said to be logarithmic if fw ( $p and df , ) ( $p+1 for each local equationf of D. A natural problem is to find the class of divisors D + X for whichiD : $•(log D) & $•((D) is a quasi-isomorphism (i.e. iD induces an isomorphism oncohomology). By Grothendieck’s comparison theorem we know that the complexes$•((D) and Rj$(C) are naturally quasi-isomorphic, where j : U = X \ D & X isthe inclusion. So, if iD is a quasi-isomorphism we say that the logarithmic compar-ison theorem holds for D (or simply LCT holds for D). In [Castro-Mond-Narvaez,Cohomology of the complement of a free divisor. Trans. Amer. Math. Soc. 348(8),3037–3049, 1996] it is proven that if D + X is a locally quasi-homogeneous freedivisor then LCT holds for D.

The reciprocal of the previous result is proved for dimension 2 in [Calderon-Mond-Narvaez-Castro, Logarithmic Cohomology of the Complement of a PlaneCurve. Comment. Math. Helv. 77(1), 24–38, 2002].

On the other hand, in [Ucha, Ph. D. Thesis, Universidad de Sevilla, 1999] itis considered the Dp-module 'M log f := Dp/(I log f where (I log f is the left ideal of Dp

1Departamento de AlgebraUniversidad de Sevilla, Spain, [email protected], [email protected]

43

generated by the linear di!erential operators of order 1 that annihilate the meromor-phic function 1/f . There exists a natural morphism %f : 'M log f & O[(D]p definedby %D(P ) = P (1/f). It is another natural question to ask for the class of D suchthat the morphism %f is an isomorphism. For Spencer free divisors, this condition isequivalent to LCT (see [Castro-Ucha, Testing the Logarithmic Comparison Theoremfor Spencer free divisors. Exp. Math., 13(4), 441-449, 2004]).

It is a work in progress to extend some of the results of the free case to the largerclass of quasi free divisors. A germ of divisor (D, 0) in (Cn, 0) is called quasi free ifthere exists a O-submodule %(D) + Der(! log D) of rank n verifying:a) %(D) is a Lie subalgebra of Der(! log D).b) There exists a basis #1, . . . , #n of %(D) such that if #i =

!j aij*j then det((aij))

is an (non necessarily reduced) equation of (D, 0).c) D%(D) = DDer(! log D).

It is an open problem to provide a e!ective criterion to decide if a given divisor isquasi free (for the free case, we have Saito’s criterion [loc. cit. Saito, 1980] which ise!ective). The aim of our talk is to present some interesting examples of these quasifree divisors that naturally appears in the context of arrangements of hyperplanes[Orlik-Terao, Arrangements and Milnor fiber. Math. Ann. 301, pp. 211-235, 1995]and linear divisors [Granger-Mond-Nieto-Schulze, Linear free divisors and the globallogarithmic comparison theorem. Ann. Inst. Fourier 59, no. 2, 811–850. 2009]. Ourcomputational approach uses Macaulay 2 and Singular.

The authors are grateful to Jorge Martın-Morales for his help using the Singularlibrary dmod.lib to treat some examples.

44

Fast polynomial reduction

N.N. Vassiliev1

Abstract

The algorithm of polynomial reduction is one of the key points of any algo-rithm for finding Grobner bases from Buchberger’s one up to Grobner walk [1] orFaugere’s algorithms [2]. Standard algorithm for polynomial reduction eliminatesleading terms of polynomials step by step using leading terms of polynomials froma Grobner basis while it is possible. During this process the number of terms in theintermediate polynomials can be very huge even in the case if the result is zero.

We suggest an algorithm for polynomial reduction which uses fast evaluation ofthe images of polynomials in the quotient algebra of an ideal. The algorithm isbased on some generalization of Horner’s method for the case of sparse multivariatepolynomials [3,4]. It uses a special representation of polynomials by marked trees.The vertices of the trees are marked by coe"cients of polynomials and the edges bysome monomials, which are defined recursively when the tree constructs. The num-ber of di!erent such monomials is much less than the number of terms of polynomialcorresponded the tree. When the tree is constructed we associate multiplication op-erators in the quotient algebra to the monomials used as marks for edges. Theseoperators evaluates recursively also using monomial basis of the quotient algebra.After all we evaluate the result using algorithm with closed to minimal value ofnumber of multiplications in the quotient algebra. All the intermediate results arepresented by their normal forms therefore the complexity of this algorithm is muchmore less than of standard one. We give also upper bound of the complexity in thecase of zero dimensional ideals. This algorithm was imlemented in C package byA.Kliskunov. In this implementation it works from 100 up to 10000 times fasterthan normalization procedure implemented in FGb package in Maple.

[1] Collart, S., Kalkbrener, M., and Mall, D., Converting bases with the Grobnerwalk, J. Symbolic Computation 24, 465-469, 1997.

[2] Faugere, J.C., A new e"cient algorithm for computing Grobner bases (f4),Journal of Pure and Applied Algebra 139, 61-88, 1999.

[3] Vassiliev, N.N., Complexity of monomial evaluations and duality. In: Com-puter Algebra in Scientifig Computing, V.G. Ganzha et al (Eds.) 479-485,Springer, 1999.

1St. Petersburg department of Steklov Institute of Mathematics, St. Petersburg, Russia

45

[4] Vassiliev, N.N., Joint evaluations of sparse polynomials and the synthesis ofevaluating programms. In: Proceedings of International Conf. Systems andMethods of Analytical Computations with Computers in Theor. Phys. Dubna,1985, pp. 154-159, (In Russian).

46

Di!erential polynomials and

standard bases

A. I. Zobnin1

Abstract

Di!erential Grobner bases (or Di!erential standard bases, DSB for short) wereintroduced independently by G. Carra Ferro and F. Ollivier in late 1980-s. Theyare natural generalizations of Grobner bases to the case of di!erential ideals.

Consider a di!erential polynomial ring F{ y} over a field of constants F of char-acteristic zero. The elements of this ring correspond to autonomous di!erentialequations in one unknown function y. The algebraic study of systems of such dif-ferential equations leads to problems concerning di!erential ideals in F{y}. Themembership problem is the classical one. It is undecidable for infinitely generateddi!erential ideals in general. On the other hand, for finitely generated ideals it is stillopen. There are beautiful algorithmic solution to this problem in the case of radicalideals via decomposition of a radical ideal into finite intersection of charactarizable(or regular) components. Besides, one can algorithmically check whether a givenpolynomial belongs to an isobaric ideal. We consider the third case, the case of ide-als that have finite (or parametric) di!erential standard basis w.r.t. some admissibleordering. The di!erence between DSBs and characteristic sets lies in the reductionprocess. While characteristic sets technique involves pseudo-reduction, DSBs needfull di!erential reduction (without initials and separants). As a result, DSBs maybe infinite even in very simple cases, but they can be applied to non-radical ideals.

The interest in di!erential standard bases reappeared after new examples offinite and recursive bases w.r.t. more general admissible orderings had been foundby the author. In fact the existence of finite DSB of an ideal depends on admissibleordering. We extended the notion of admissible ordering on di!erential monomialsand introduced special classes of orderings that share common properties w.r.t.di!erential operations (e.g., strict #-stable orderings, #-lexicographical orderings, '-orderings and so on). It turned out that finiteness conditions of DSBs are closelyrelated with existence of linear and quasi-linear polynomials (or their powers) in theideal. Moreover, we found a link between existence of finite DSB and property ofbeing radical for ideals generated by at most first-order polynomial.

The study of finiteness criteria for di!erential standard bases for some classes oforderings is not completed yet, but there are some conjectures that are proved in

1Faculty of Mechanics and Mathematics, Moscow State University, Russia,[email protected]

47

most cases. We present the overview of the theory of di!erential standard bases andalso state some open combinatorial problems concerning di!erential polynomials(for example, whether it is possible to check algorithmically if a di!erential idealcontains a quasi-linear polynomial).

[1] Carra Ferro, G., Groebner bases and di!erential algebra, Lecture Notes inComputer Science 356, 141-150, 1989.

[2] Carra Ferro, G., A survey on di!erential Grobner bases. In: M. Rosenkranzand D. Wang (Eds.), Grobner Bases in Symbolic Analysis, Proc. SpecialSemester on Grobner Bases and Related Methods, de Gruyter, 2007.

[3] Ollivier, F., Standard bases of di!erential ideals, Lecture Notes in ComputerScience 508, 304-321, 1990.

[4] Zobnin, A.I., Admissible orderings and finiteness criteria for di!erential stan-dard bases, In: Proceedings of ISSAC-2005, 2005, pp. 365-372.

Participants’ addresses

51

List of participants

Abad, AlbertoDpt. Fısica Teorica, GME and IUMA, University of Zaragoza (Spain),[email protected]

Algaba, AntonioDpt. Matematicas, University of Huelva (Spain), [email protected]

Barrio, RobertoDpt. Matematica Aplicada, GME and IUMA, University of Zaragoza (Spain),[email protected]

Blinkov, YuriDpt. Mathematics and Mechanics, Saratov State University (Russia),[email protected]

Castro-Jimenez, Francisco J.Dpt. Algebra, University of Sevilla (Spain), [email protected]

Crespo, FranciscoDpt. Matematica Aplicada, University of Murcia (Spain),[email protected]

Edneral, Victor F.Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University(Russia), [email protected]

Efimovskaya, Olga V.Laboratory of Computational Methods, Faculty of Mechanics andMathematics, Lomonosov Moscow State University (Russia),[email protected]

Elipe, AntonioDpto. Matematica Aplicada, GME and IUMA, University of Zaragoza (Spain),[email protected]

Evlakhov, Simeon A.Laboratory of Information Technologies, Joint Institute for Nuclear Research,Dubna (Russia), [email protected]

52

Garcıa, CristobalDpto. Matematicas, University of Huelva (Spain), [email protected]

Gerdt, Vladimir P.Group of Algebraic and Quantum Computations, Joint Institute for NuclearResearch, Dubna (Russia), [email protected]

Gusev, Alexander A.Laboratory of Information Technologies, Joint Institute for Nuclear Research,Dubna (Russia), [email protected]

Hernando, AntonioDpt. Sistemas Inteligentes Aplicados, Polytechnic University of Madrid (Spain),[email protected]

Higueras, InmaculadaDpt. Ingenierıa Matematica e Informatica, State University of Navarre, Pam-plona (Spain), [email protected]

Kornyak, VladimirLaboratory of Information Technologies, Joint Institute for Nuclear Research,Dubna (Russia), [email protected]

Lanchares, VıctorDpt. Matematicas y Computacion, University of La Rioja, Logrono (Spain),[email protected]

Lara, MartınSpanish Navy Observatory, San Fernando, Cadiz (Spain), [email protected]

Morales-Ruiz, Juan J.Dep. Matematica Aplicada a la Ingenierıa Civil, Polytechnic University ofMadrid (Spain), [email protected]

Myllari, AleksandrDepartment of Physics and Astronomy, University of Turku (Finland),[email protected]

Palacian, Jesus F.Dpt. Ingenierıa Matematica e Informatica, State University of Navarre, Pam-plona (Spain), [email protected]

Pascual, Ana I.Dpt. Matematicas y Computacion, University of La Rioja, Logrono (Spain),[email protected]

53

Reyes, ManuelDpto. Matematicas, University of Huelva (Spain), [email protected]

Rodrıguez, MarcosDpto. Matematica Aplicada, GME and IUMA, University of Zaragoza (Spain),[email protected]

Salas, J. PabloArea de Fısica Aplicada, University of La Rioja, Logrono (Spain),[email protected]

San Juan, Juan FelixDpt. Matematicas y Computacion, University of La Rioja, Logrono (Spain),[email protected]

Sansottera, MarcoDip. Matematica, The University of Milan (Italy),[email protected]

Stefanescu, DoruDip. Mathematics, University of Bucharest (Romania),[email protected]

Sturm, ThomasDpt. Matematicas, Estadıstica y Computacion, Universidad de Cantabria,Santander (Spain), [email protected]

Ucha, Jose MarıaDpt. Algebra, University of Sevilla (Spain), [email protected]

Vassiliev, Nikolay N.Dpt. Steklov Institute of Mathematics, St. Petersburg (Russia),[email protected]

Yanguas, PatriciaDpt. Ingenierıa Matematica e Informatica, State University of Navarre, Pam-plona (Spain), [email protected]

Zobnin, AlexeyFaculty of Mechanics and Mathematics, Moscow State University (Russia),[email protected]

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