Solving polynomial equations. Foundations, algorithms, and applications

436 Pages·2005·2.73 MB·English

by  Alicia Dickenstein

Algorithms and Computation
in Mathematics • Volume 14
Editors
ManuelBronstein ArjehM.Cohen
HenriCohen DavidEisenbud
BerndSturmfels
Alicia Dickenstein
Ioannis Z. Emiris (Editors)
Solving Polynomial
Equations
Foundations, Algorithms, and Applications
With46Figures
123
Editors
AliciaDickenstein
DepartamentodeMatem´atica
FacultaddeCienciasExactasyNaturales
UniversidaddeBuenosAires
CiudadUniversitaria–Pab.I
C1428EGABuenosAires,Argentina
e-mail:[email protected]
IoannisZ.Emiris
DepartmentofInformaticsandTelecommunications
NationalKapodistrianUniversityofAthens
PanepistimiopolisGR-15784,Greece
e-mail:[email protected]
LibraryofCongressControlNumber:2005921469
MathematicsSubjectClassification(2000):14QXX,68W30,65H10,13PXX,11C08,
12F10,12Y05,13P05,13P10,32A27,52A39,52B11,52B20,68Q25,68U07
ISSN1431-1550
ISBN-10 3-540-24326-7 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-24326-7 SpringerBerlinHeidelbergNewYork
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Preface
The subject of this book is the solution of polynomial equations, that is, sys-
tems of (generally) non-linear algebraic equations. This study is at the heart
of several areas of mathematics and its applications. It has provided the mo-
tivation for advances in different branches of mathematics such as algebra,
geometry, topology, and numerical analysis. In recent years, an explosive de-
velopment of algorithms and software has made it possible to solve many
problems which had been intractable up to then and greatly expanded the
areas of applications to include robotics, machine vision, signal processing,
structuralmolecularbiology,computer-aideddesignandgeometricmodelling,
as well as certain areas of statistics, optimization and game theory, and bio-
logical networks. At the same time, symbolic computation has proved to be
an invaluable tool for experimentation and conjecture in pure mathematics.
As a consequence, the interest in effective algebraic geometry and computer
algebrahasextendedwellbeyonditsoriginalconstituencyofpureandapplied
mathematicians and computer scientists, to encompass many other scientists
and engineers. While the core of the subject remains algebraic geometry, it
also calls upon many other aspects of mathematics and theoretical computer
science, ranging from numerical methods, differential equations and number
theory to discrete geometry, combinatorics and complexity theory.
Thegoalofthisbookistoprovideageneralintroductiontomodernmath-
ematical aspects in computing with multivariate polynomials and in solving
algebraicsystems.Itisaimedtoupper-levelundergraduateandgraduatestu-
dents, and researchers in pure and applied mathematics and engineering, in-
terested in computational algebra and in the connections between computer
algebra and numerical mathematics. Most chapters assume a solid ground-
ing in linear algebra while for several of them a basic knowledge of Gro¨bner
bases, at the level of [CLO97] is expected. Gr¨obner bases have become a ba-
sic standard tool in computer algebra and the reader may consult any other
textbook such as [AL94, BW93, CLO98, GP02], or the introductory chapter
in [CCS99]. Below we discuss briefly the content of each chapter and some of
their prerequisites.
VIII Preface
The book describes foundations, recent developments and applications of
Gro¨bner and border bases, residues, multivariate resultants, including toric
elimination theory, primary decomposition of ideals, multivariate polynomial
factorization, as well as homotopy continuation methods. While some of the
chaptersareintroductoryinnature,otherspresentthestate-of-the-artinsym-
bolic techniques in polynomial system solving, including effective and algo-
rithmicmethodsinalgebraicgeometryandcomputationalalgebra,complexity
issues,andapplications.Wealsodiscussseveralnumericandsymbolic-numeric
methods.Thisisnotastandardtextbookinthateachchapterisindependent
and, largely, self-contained. However, there are strong links between the dif-
ferent chapters as evidenced by the many cross-references. While the reader
gains the advantage of being able to access the book at many different places
and of seeing the interplay of different views of the same concepts, we should
note that, because of the different needs and traditions, some notations in-
evitablyvarybetweendifferentchapters.Wehavetriedtonotethisinthetext
whenever it occurs. The single bibliography and index underline the unity of
the subject.
The first chapter gives an introduction to the notions of residues and re-
sultants, and the interplay between them, starting with the univariate case
andsynthesizingdifferentapproaches.Thesectionsonunivariateresiduesand
resultants could be used in an undergraduate course on complex analysis, ab-
stractalgebra,orcomputationalalgebraasanintroductiontomoreadvanced
topicsandtoillustratetheinterdependenceofdifferentareasofmathematics.
The multivariate sections, on the other hand, directed to graduate students
andresearchers,areintendedasanintroductiontoconceptswhicharewidely
used in current research and applications.
The second chapter puts the accent on linear algebra methods to deal
with polynomial systems: the multiplication maps in the quotient algebra by
a polynomial ideal are linear and allow for the use of eigenvalues and eigen-
vectors, duality, etc. Applications to Galois theory, factoring, and primary
decomposition are offered. The first sections require, besides standard linear
algebra,somebackgroundoncomputationalalgebraicgeometry(forinstance,
the first five chapters of [CLO97]). Some acquaintance with local rings (as in
Chapter 4 of [CLO98]) would also be helpful. Known basic facts about field
extensions and Galois theory are assumed in the last part.
Thethirdchapteralsoelaboratesontheconceptsinthefirsttwochapters,
and combines them with numerical methods for polynomial system solving
and several applications. The tools and methods developed are used to solve
problems arising in implicitization of rational surfaces, determination of the
position of a camera or a parallel robot, molecular conformations, and blind
identification in signal processing. The required background is very similar to
that needed for the first sections of Chapter 2.
Chapter4isdevotedtolayingthealgebraicfoundationsforborderbasesof
ideals,anextensionofthetheoryofGro¨bnerbases,yieldingmoreflexiblebases
of the quotient algebras. Border bases yield a connection between computer
Preface IX
algebra and numerical mathematics. An application to design of experiments
in statistics is included.
The fifth chapter concentrates on various techniques for computing pri-
marydecomposition of ideals.This machinery isapplied tostudyan interest-
ing class of ideals coming from Bayesian networks, establishing an important
linkbetweenalgebraicgeometryandtheemergingfieldofalgebraicstatistics.
BesidesGr¨obnerbases,thereadersareexpectedtohaveacasualunderstand-
ing of the algebra-geometry dictionary between ideals in polynomial rings
and their zero set. Many propositions that can be found in the literature
arestated without proof, but thechapter contains several accessible exercises
dealing with the structure and decomposition of polynomial ideals.
Chapter 6 studies the inherent complexity of polynomial system solving
when working with the dense encoding of the input polynomials and under
the model of straight-line programs, i.e., when polynomials are not given by
their monomials but by evaluation programs. Being a brief survey of alge-
braic complexity applied to computational algebraic geometry, there is not
much background required, though knowledge of basic notions of algebraic
geometry and commutative algebra would be helpful. The chapter is mostly
self-contained; when necessary, basic bibliography supplements are indicated.
Chapter 7 is devoted to the study of sparse systems of polynomial equa-
tions, i.e., algebraic equations with a specific monomial structure, presenting
a comprehensive state-of-the-art introduction to the field. Combinatorial and
discrete geometry, together with matrices of special structure, are ingredients
of the presentation of toric (or sparse) elimination theory. The chapter fo-
cuses on applications to geometric modelling and computer-aided design. It
also provides the tools for exploiting the structure of algebraic systems which
mayariseindifferentapplications.Somebasicknowledgeofdiscretegeometry
for polyhedral objects in arbitrary dimension is assumed. This chapter will
be of particular interest to graduate students and researchers in theoretical
computer science or applied mathematics wishing to combine discrete and
algebraic geometry.
Chapter 8 deals with numerical algebraic geometry, a term coined some
years ago to describe a new field, which bears the same relation to algebraic
geometryasnumericallinearalgebradoestolinearalgebra.Modernhomotopy
methodstodescribesolutioncomponentsofpolynomialsystemsarepresented.
The prerequisites include a basic course in numerical analysis, in particular
Newton’s method for nonlinear systems. Because of the numerical flavor of
the proposed methods, this chapter is expected to be particularly appealing
to engineers.
Lastly,Chapter9givesacompleteoverviewofoldandrecentmethodsfor
theimportantproblemofapproximatefactorizationofamultivariatepolyno-
mial,inotherwords,thecomplexirreducibledecompositionofahypersurface.
The main techniques rely on approximate numerical computation but the re-
sults are exact and certified. It is addressed to students and researchers with
X Preface
some basic knowledge of commutative algebra, algebraic numbers and holo-
morphic functions of several variables.
This book grew out of Course Notes prepared for the CIMPA Graduate
SchoolonSystemsofPolynomialEquationthatweorganizedinBuenosAires,
in July 2003. We take this opportunity to thank CIMPA for the funding and
the academic support to carry out this activity. We are also grateful for the
support from the following institutions: International Centre for Theoreti-
cal Physics (ICTP, Italy), Consejo Nacional de Investigaciones Cient´ıficas y
T´ecnicas(CONICET,Argentina),InstitutNationaldeRechercheenInforma-
tique et en Automatique (INRIA, France), PROSUL Programme from CNPq
(Brazil), D´el´egation r´egionale de coop´eration Franc¸aise au Chili, and Univer-
sidad de Buenos Aires (Argentina). We also thank ECOS-Sud, whose project
A00E02 between INRIA and Universidad de Buenos Aires provided the ini-
tialframeworkforourcollaboration.SpecialthanksgotoGregorioMalajovich
and Alvaro Rittatore, who co-organized with us the I Latin American Work-
shoponPolynomialSystemswhichfollowedtheSchool.Finally,wewouldlike
to thank deeply all the speakers and all the participants.
December 2004 Alicia Dickenstein and Ioannis Z. Emiris
Contents
1 Introduction to residues and resultants
Eduardo Cattani, Alicia Dickenstein................................ 1
1.0 Introduction ................................................. 1
1.1 Residues in one variable....................................... 3
1.2 Some applications of residues .................................. 8
1.3 Resultants in one variable ..................................... 16
1.4 Some applications of resultants................................. 21
1.5 Multidimensional residues ..................................... 27
1.6 Multivariate resultants ........................................ 44
1.7 Residues and resultants ....................................... 55
2 Solving equations via algebras
David A. Cox ................................................... 63
2.0 Introduction ................................................. 63
2.1 Solving equations............................................. 64
2.2 Ideals defined by linear conditions .............................. 78
2.3 Resultants................................................... 91
2.4 Factoring....................................................100
2.5 Galois theory ................................................114
3 Symbolic-numeric methods for solving polynomial equations
and applications
Mohamed Elkadi, Bernard Mourrain................................125
3.0 Introduction .................................................125
3.1 Solving polynomial systems....................................126
3.2 Structure of the quotient algebra ...............................131
3.3 Duality .....................................................141
3.4 Resultant constructions .......................................145
3.5 Geometric solvers ............................................151
3.6 Applications .................................................158
XII Contents
4 An algebraist’s view on border bases
Achim Kehrein, Martin Kreuzer, Lorenzo Robbiano...................169
4.0 Introduction .................................................169
4.1 Commuting endomorphisms ...................................172
4.2 Border prebases ..............................................179
4.3 Border bases.................................................186
4.4 Application to statistics .......................................195
5 Tools for computing primary decompositions and
applications to ideals associated to Bayesian networks
Michael Stillman.................................................203
5.0 Introduction .................................................203
5.1 Lecture 1: Algebraic varieties and components ...................205
5.2 Lecture 2: Bayesian networks and Markov ideals..................212
5.3 Lecture 3: Tools for computing primary decompositions ...........219
5.4 Lecture 4: Putting it all together ...............................228
6 Algorithms and their complexities
Juan Sabia......................................................241
6.0 Introduction and basic notation ................................241
6.1 Statement of the problems.....................................242
6.2 Algorithms and complexity ....................................247
6.3 Dense encoding and algorithms.................................248
6.4 Straight-line Program encoding for polynomials ..................255
6.5 The Newton-Hensel method ...................................263
6.6 Other trends.................................................266
7 Toric resultants and applications to geometric modelling
Ioannis Z. Emiris ................................................269
7.0 Introduction .................................................269
7.1 Toric elimination theory.......................................270
7.2 Matrix formulae..............................................279
7.3 Implicitization with base points ................................288
7.4 Implicit support..............................................292
7.5 Algebraic solving by linear algebra..............................298
8 Introduction to numerical algebraic geometry
Andrew J. Sommese, Jan Verschelde, Charles W. Wampler ...........301
8.0 Introduction .................................................302
8.1 Homotopy continuation methods – an overview...................303
8.2 Homotopies to approximate all isolated solutions .................305
8.3 Homotopies for positive dimensional solution sets.................326
8.4 Software and applications .....................................335