Computer Algebra and Polynomials: Applications of Algebra and Number Theory

222 Pages·2015·9.04 MB·English

by  Jaime Gutierrez

t Jaime Gutierrez
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Applications of Algebra and Number Theory
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Lecture Notes in Computer Science 8942
Commenced Publication in 1973
Founding and Former Series Editors:
Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen
Editorial Board
David Hutchison
Lancaster University, Lancaster, UK
Takeo Kanade
Carnegie Mellon University, Pittsburgh, PA, USA
Josef Kittler
University of Surrey, Guildford, UK
Jon M. Kleinberg
Cornell University, Ithaca, NY, USA
Friedemann Mattern
ETH Zurich, Zürich, Switzerland
John C. Mitchell
Stanford University, Stanford, CA, USA
Moni Naor
Weizmann Institute of Science, Rehovot, Israel
C. Pandu Rangan
Indian Institute of Technology, Madras, India
Bernhard Steffen
TU Dortmund University, Dortmund, Germany
Demetri Terzopoulos
University of California, Los Angeles, CA, USA
Doug Tygar
University of California, Berkeley, CA, USA
Gerhard Weikum
Max Planck Institute for Informatics, Saarbrücken, Germany
More information about this series at http://www.springer.com/series/7407
Jaime Gutierrez Josef Schicho
(cid:129)
Martin Weimann (Eds.)
Computer Algebra
and Polynomials
Applications of Algebra and Number Theory
123
Editors
JaimeGutierrez MartinWeimann
Universityof Cantabria Universityof Caen
Santander Caen
Spain France
JosefSchicho
Ricam Linz
Linz
Austria
ISSN 0302-9743 ISSN 1611-3349 (electronic)
Lecture Notesin ComputerScience
ISBN 978-3-319-15080-2 ISBN 978-3-319-15081-9 (eBook)
DOI 10.1007/978-3-319-15081-9
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Preface
This textbook regroups selected papers of the Workshop on Computer Algebra and
Polynomials,whichwas held inLinzattheJohann RadonInstitute for Computational
and Applied Mathematics (RICAM) during November 25–29, 2013, on the occasion
of the Special Semester on Applications of Algebra and Number Theory. The work-
shopincludedinvitedtalksandcontributedtalks.Authorsofselectedcontributedtalks
were invited to submit a paper to these proceedings.
This workshop focuses on the theory and algorithms for polynomials over various
coefficientdomainsthatmaybe(butisnotrestrictedto)acommutativealgebra,suchas
a finite field or ring. The operations on polynomials in the focus are factorization,
composition and decomposition, basis computation for modules, etc. Algorithms for
such operations on polynomials have always been of central interest in computer
algebra,asitcombinesformal(thevariables)andalgebraicornumeric(thecoefficients)
aspects.
The plan wastobring togetheramix ofexpertsfor thevariouscoefficient domains
in order to explore similarities as well as differences. Also experts for applications of
manipulation of polynomials were invited, such as polynomial system solving or the
analysis of algebraic varieties.
The workshop contributions were selected through a rigorous reviewing process
based on anonymous reviews made by various expert reviewers. There were usually
two reviewers for one submission. The process was simple blind as authors did not
know the names of the reviewers evaluating their papers. We have chosen 12 articles
fromthemanyexcellentsubmissionswereceived.Wehopethatthereaderwillfindan
interesting perspective of this rich and active area. Let us mention here a few words
about each of the selected papers.
The expository paper by Felix Breuer gives an introduction to Ehrhart theory and
takesatourthroughitsapplicationsinenumerativecombinatorics.ThepaperbyCarlos
D’Andrea presents several methods and open questions for dealing in a more efficient
waywiththeimplicitizationofrationalparameterization.ThesurveypaperbyJoachim
von zur Gathen and Konstantin Ziegler presents several counting results for inde-
composable/decomposable polynomials over finite fields. Willem A. de Graaf
describes methods for dealing with the problem of deciding whether a given element
ofthevector space liesintheclosureoftheorbitofanothergiven element.The paper
by Georg Grasegger and Franz Winkler presents a new and rather general method for
solving algebraic ordinary differential equations. The paper by Manuel Kauers, Max-
imilian Jaroschek, and Fredrik Johansson shows a Sage implementation of Ore alge-
bras. The paper by Zoltán Kovács and Bernard Parisse presents several changes for
solvingequationsystemoftheGeoGebrasoftware.RagniPiene’spaperstudiesseveral
concepts of the classical polar varieties. The paper by Cristian-Silviu Radu solves an
open problem about modular polynomials of levels 3 and 5. The survey paper by
Carsten Schneider presents algorithms and their efficiency for some parameterized
VI Preface
telescoping problems. The paper by J. Rafael Sendra, David Sevilla, and Carlos
Villarino provides sufficient conditions for a parameterization to be surjective and
computingasetofthepointsnotcoveredbytheparameterization.Finally,thepaperby
Maria-Laura Torrente presents an overview of the problem of the representation of
rationalsurfaceassettheoreticcompleteintersectionandalsoanoriginalproofthatthe
rational normal quartic is set-theoretically complete intersection of quadrics.
Allacceptedpapers,exceptone,werepresentedattheworkshopduringtalksof25
or45min.Therewerealsosometalkswithnocontributiontotheproceedings,seethe
website http://www.ricam.oeaw.ac.at/specsem/specsem2013/workshop3/ of the work-
shop for the list of all speakers and abstracts.
We would like to thank all the speakers for their contributions to the program, and
alltheauthorswhohavesubmittedtheirpreciousmanuscriptstothisbook.Wewould
also like to ask for their understanding for our possible mistakes. The workshop and
this volume would not have been possible without the contributions of numerous
individuals and organizations, and we sincerely thank them for their support.
November 2014 Jaime Gutierrez
Josef Schicho
Martin Weimann
Organization
Organizers and Scientific Committee
Jaime Gutierrez University of Cantabria, Spain
Josef Schicho Johann Radon Institute, Austria
Martin Weimann University of Caen, France
Contents
An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications
in Enumerative Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Felix Breuer
Moving Curve Ideals of Rational Plane Parametrizations . . . . . . . . . . . . . . . 30
Carlos D’Andrea
Survey on Counting Special Types of Polynomials . . . . . . . . . . . . . . . . . . . 50
Joachim von zur Gathen and Konstantin Ziegler
Orbit Closures of Linear Algebraic Groups. . . . . . . . . . . . . . . . . . . . . . . . . 76
Willem A. de Graaf
Symbolic Solutions of First-Order Algebraic ODEs. . . . . . . . . . . . . . . . . . . 94
Georg Grasegger and Franz Winkler
Ore Polynomials in Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Manuel Kauers, Maximilian Jaroschek, and Fredrik Johansson
Giac and GeoGebra – Improved Gröbner Basis Computations . . . . . . . . . . . 126
Zoltán Kovács and Bernard Parisse
Polar Varieties Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Ragni Piene
A Note on a Problem Proposed by Kim and Lisonek. . . . . . . . . . . . . . . . . . 151
Cristian-Silviu Radu
Fast Algorithms for Refined Parameterized Telescoping in Difference
Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Carsten Schneider
Some Results on the Surjectivity of Surface Parametrizations. . . . . . . . . . . . 192
J. Rafael Sendra, David Sevilla, and Carlos Villarino
Rational Normal Curves as Set-Theoretic Complete Intersections
of Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Maria-Laura Torrente
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
An Invitation to Ehrhart Theory:
Polyhedral Geometry and its Applications
in Enumerative Combinatorics
B
Felix Breuer( )
Research Institute for Symbolic Computation, Johannes Kepler University,
Altenberger Str. 69, 4040 Linz, Austria
[e-mail]
http://www.felixbreuer.net
Abstract. InthisexpositoryarticlewegiveanintroductiontoEhrhart
theory, i.e., the theory of integer points in polyhedra, and take a tour
through its applications in enumerative combinatorics. Topics include
geometricmodelingincombinatorics,Ehrhart’smethodforprovingthat
a counting function is a polynomial, the connection between polyhe-
dral cones, rational functions and quasisymmetric functions, methods
forboundingcoefficients,combinatorialreciprocitytheorems,algorithms
forcountingintegerpointsinpolyhedraandcomputingrationalfunction
representations,aswellasvisualizationsofthegreatestcommondivisor
and the Euclidean algorithm.
· · ·
Keywords: Polynomial Quasipolynomial Rational function Qua-
· · ·
sisymmetric function Partial polytopal complex Simplicial cone
· ·
Fundamentalparallelepiped Combinatorialreciprocitytheorem Barvi-
· · ·
nok’salgorithm Euclideanalgorithm Greatestcommondivisor Gen-
· ·
erating function Formal power series Integer linear programming
1 Introduction
Polyhedralgeometryisapowerfultoolformakingthestructureunderlyingmany
combinatorialproblemsvisible–oftenliterally!Inthisexpositoryarticlewegive
anintroductiontoEhrharttheoryandmoregenerallythetheoryofintegerpoints
in polyhedra and take a tour through some of its many applications, especially
in enumerative combinatorics.
In Sect.2, we start with two classic examples of geometric modeling in com-
binatorics and then introduce Ehrhart’s method for showing that a counting
function is a (quasi-)polynomial in Sect.3. Wepresentcombinatorial reciprocity
theoremsasafirstapplicationinSect.4,beforewetalkaboutconesasthebasic
building block of Ehrhart theory in Sect.5. The connection of cones to rational
FelixBreuerwassupportedbyAustrianScienceFund(FWF)specialresearchgroup
Algorithmic and Enumerative Combinatorics SFB F50-06.
(cid:2)c SpringerInternationalPublishingSwitzerland2015
J.Gutierrezetal.(Eds.):ComputerAlgebraandPolynomials,LNCS8942,pp.1–29,2015.
DOI:10.1007/978-3-319-15081-91