Elimination methods in polynomial algebra

248 Pages·1998·1.252 MB·English

by  V. Bykov, A. Kytmanov, M. Lazman, Mikael Passare

Elimination Methods in
Polynomial Computer Algebra

Valery Bykov
Alexander Kytmanov
Mark Lazman
ELIMINATION METHODS
IN POLYNOMIAL COMPUTER
ALGEBRA
Kluwer Academic Publishers
Eliminationmethods in polynomialcomputer algebra(cid:0) Valery Bykov(cid:1) Alexander
Kytmanov(cid:1) Mark Lazman(cid:2) (cid:3) Kluwer Academic Publishers(cid:1) (cid:4)(cid:5)(cid:5)(cid:6)(cid:2)
The bookpresents amodi(cid:7)edmethod(cid:1)based onmultidimensionalresidue theory(cid:1)
for the eliminationof unknowns from a system of nonlinear algebraic equations(cid:2) An
algorithm is given for constructing the resultant of the system(cid:1) and a computer
implementation making use of formula manipulation software is carried out(cid:2) The
texts of the worked out programs in the system Maple are provided(cid:2)
The obtained algorithms and programs are then applied to questions from the
theory of chemical kinetics(cid:1) such as the search for all stationary solutions of kinetic
equations and the construction of kinetic polynomials(cid:2)
The subject of this book is closely connected with a wide range of current prob(cid:8)
lems in the analysis of nonlinear systems(cid:2)
Table of Contents
INTRODUCTION vii
PREFACE TO THE ENGLISH EDITION ix
PREFACE BY THE ENGLISH EDITOR x
LIST OF SYMBOLS xi
Chapter (cid:4)(cid:2) BASIC MATHEMATICAL FACTS (cid:4)
(cid:4)(cid:2) The logarithmic residue (cid:4)
(cid:9)(cid:2) The Newton recursion formulas (cid:10)
(cid:11)(cid:2) Localization theorems for the real zeros of a polynomial (cid:4)(cid:4)
(cid:12)(cid:2) The local residue (cid:13)of Grothendieck(cid:14) (cid:4)(cid:10)
(cid:15)(cid:2) The multidimensional logarithmic residue (cid:9)(cid:4)
(cid:10)(cid:2) The classical scheme for elimination of unknowns (cid:9)(cid:12)
Chapter (cid:9)(cid:2) A MODIFIED ELIMINATION METHOD (cid:11)(cid:4)
(cid:6)(cid:2) A generalized transformation formula for local residues (cid:11)(cid:4)
(cid:16)(cid:2) A modi(cid:7)ed elimination method (cid:11)(cid:15)
(cid:5)(cid:2) A formula for the logarithmic derivative of the resultant (cid:10)(cid:11)
(cid:4)(cid:17)(cid:2) Multidimensional analogues of the Newton formulas (cid:6)(cid:16)
(cid:4)(cid:4)(cid:2) Elimination of unknowns in di(cid:18)erent variables(cid:2) Real roots (cid:16)(cid:16)
Chapter (cid:11)(cid:2) APPLICATIONS IN MATHEMATICAL KINETICS (cid:5)(cid:5)
(cid:4)(cid:9)(cid:2) Short schemes (cid:4)(cid:17)(cid:4)
(cid:4)(cid:9)(cid:2)(cid:4)(cid:2) The parallel scheme (cid:4)(cid:17)(cid:4)
(cid:4)(cid:9)(cid:2)(cid:9)(cid:2) The sequential scheme (cid:4)(cid:4)(cid:10)
(cid:4)(cid:11)(cid:2) The search for all stationary solutions (cid:4)(cid:9)(cid:6)
(cid:4)(cid:12)(cid:2) The kinetic polynomial(cid:2) Single(cid:8)route mechanisms (cid:4)(cid:11)(cid:6)
(cid:4)(cid:15)(cid:2) Construction of the kinetic polynomial in the general case (cid:4)(cid:10)(cid:9)
Chapter (cid:12)(cid:2) COMPUTER REALIZATIONS (cid:4)(cid:6)(cid:11)
(cid:4)(cid:10)(cid:2) Analytic manipulations on the computer (cid:4)(cid:6)(cid:11)
(cid:4)(cid:10)(cid:2)(cid:4)(cid:2) Special features(cid:1) (cid:7)eld of applications (cid:4)(cid:6)(cid:11)
(cid:4)(cid:10)(cid:2)(cid:9)(cid:2) The system Maple (cid:4)(cid:6)(cid:15)
(cid:4)(cid:6)(cid:2) Basic problems in computer algebra of polynomials (cid:4)(cid:16)(cid:17)
(cid:4)(cid:6)(cid:2)(cid:4)(cid:2) Applied problems and polynomial algebra (cid:4)(cid:16)(cid:17)
(cid:4)(cid:6)(cid:2)(cid:9)(cid:2) Methods and algorithms (cid:4)(cid:16)(cid:12)
(cid:4)(cid:6)(cid:2)(cid:9)(cid:2)(cid:4)(cid:2) The resultant of two polynomials (cid:4)(cid:16)(cid:12)
(cid:4)(cid:6)(cid:2)(cid:9)(cid:2)(cid:9)(cid:2) Algorithms for elimination of unknowns (cid:4)(cid:16)(cid:15)
v
vi TABLE OF CONTENTS
(cid:4)(cid:6)(cid:2)(cid:9)(cid:2)(cid:11)(cid:2) The algorithms of solvability theory (cid:4)(cid:5)(cid:4)
(cid:4)(cid:16)(cid:2) Realization of the elimination method (cid:4)(cid:5)(cid:11)
(cid:4)(cid:16)(cid:2)(cid:4)(cid:2) Calculation of the resultant of two polynomials (cid:4)(cid:5)(cid:11)
(cid:4)(cid:16)(cid:2)(cid:9)(cid:2) Conditions for non(cid:19)uniqueness of steady(cid:19)states (cid:4)(cid:5)(cid:15)
(cid:4)(cid:16)(cid:2)(cid:11)(cid:2) Realization of the method of Gr(cid:20)obner bases (cid:4)(cid:5)(cid:5)
(cid:4)(cid:5)(cid:2) The construction of the resultant (cid:9)(cid:17)(cid:6)
(cid:4)(cid:5)(cid:2)(cid:4)(cid:2) Systems of type (cid:13)(cid:16)(cid:2)(cid:9)(cid:14) (cid:9)(cid:17)(cid:6)
(cid:4)(cid:5)(cid:2)(cid:9)(cid:2) Kinetic polynomials(cid:1) resultants and Gr(cid:20)obner bases (cid:9)(cid:4)(cid:15)
(cid:4)(cid:5)(cid:2)(cid:11)(cid:2) A program for calculating the kinetic polynomial (cid:9)(cid:4)(cid:16)
List of applications (cid:9)(cid:9)(cid:15)
Bibliography (cid:9)(cid:9)(cid:6)
Index (cid:9)(cid:11)(cid:15)
vii
INTRODUCTION
The subject ofthisbookisconnectedwithanewdirectioninmathematics(cid:1)which
has been actively developed over the last few years(cid:1) namely the (cid:7)eld of polynomial
computer algebra(cid:1) which lies at the intersection point of algebra(cid:1) mathematical
analysis and programming(cid:2)
There were several incentives towritethe book(cid:2) Firstof all(cid:1)there has latelybeen
a considerable interest in applied nonlinear problems characterized by multiple sta(cid:8)
tionary states(cid:2) Practical needs have then in their turn led to the appearance of new
theoretical results in the analysis of systems of nonlinear algebraic equations(cid:2) And
(cid:7)nally(cid:1) the introduction of various computer packages for analytic manipulations
has made it possible to use complicated elimination(cid:8)theoretical algorithms in prac(cid:8)
tical research(cid:2) The structure of the book is accordingly represented by three main
parts(cid:21) Mathematical results driven to constructive algorithms(cid:1) computer algebra
realizations of these algorithms(cid:1) and applications(cid:2)
Nonlinear systems of algebraic equations arise in diverse (cid:7)elds of science(cid:2) In
particular(cid:1) for processes described by systems of di(cid:18)erential equations with a poly(cid:8)
nomial right hand side one is faced with the problem of determining the number
(cid:13)and location(cid:14) of the stationary states in certain sets(cid:2) This then leads to problems
inpolynomialcomputeralgebra(cid:21) The construction ofalgorithmsfordeterminingthe
number of roots of the given equation system in di(cid:18)erent sets(cid:22) the determination of
the roots themselves(cid:22) the eliminationof some of the unknowns from the system(cid:22) the
construction of a (cid:23)triangular(cid:24) system (cid:13)Gr(cid:20)obner basis(cid:14) equivalent to the given sys(cid:8)
tem(cid:2) To deal with such questions one naturally has to develop methods for analytic
computer manipulations(cid:2)
The (cid:7)rst chapter of this monograph is of an introductory character(cid:1) and it basi(cid:8)
cally consists of a presentation of well(cid:8)known material(cid:2)
The main new mathematical results are concentrated to the second chapter(cid:2)
Here we describe a modi(cid:7)ed elimination method proposed by Aizenberg(cid:2) The idea
of this method is to use a multidimensional residue formula to express the sum of
the values of an arbitrary polynomial in the roots of a given algebraic system (cid:13)with(cid:8)
out having to (cid:7)rst determine these roots(cid:14)(cid:2) We provide formulas for (cid:7)nding such
sums for various systems of equations(cid:1) and we show how to construct intermedi(cid:8)
ate polynomials (cid:13)analogues of resultants(cid:14) for this method(cid:2) All these methods and
formulas can easily be turned into algorithms(cid:2) In particular(cid:1) for systems of three
equations the calculations are carried out completely(cid:2) A formula for (cid:7)nding the
logarithmic derivative of the resultant of a system of algebraic equations is also ob(cid:8)
tained(cid:1) and it allows one to compute the resultant itself as the sum of local residues
(cid:13)and hence to express the resultant in closed form(cid:14)(cid:2) As a corollary we get a formula
expressing the logarithmic derivative of the resultant through a sum of logarithmic
derivatives of subsystems of the given system(cid:2) These formulas allows us to write
down the coe(cid:25)cients of the resultant in (cid:7)nal form(cid:2) Multidimensional analogues of
the Newton recursion relations are given(cid:2) It turns out that there are many such
formulas(cid:1) and that they are connected with di(cid:18)erent types of systems of equations(cid:2)
Moreover(cid:1) some of the coe(cid:25)cients of these systems cannot be determined in terms
viii INTRODUCTION
of power sums of roots(cid:1) so it is necessary to consider also (cid:23)pseudopower(cid:24) sums(cid:2) Var(cid:8)
ious other problems related to the modi(cid:7)ed elimination method are also discussed(cid:21)
The determination of resultants with respect to certain groups of variables(cid:1) and the
determination of the number of real roots in domains with real analytic boundary(cid:2)
Computer realizations of the proposed methods are given in the fourth chapter(cid:2)
As our basic software for analytic manipulations we have chosen the system RE(cid:8)
DUCE(cid:2) Two important problems of polynomial computer algebra are emphasized(cid:21)
The construction of resultants and computer realizations of eliminationmethods(cid:2) In
the fairly voluminous applications provided at the end of the book we have included
the texts of the programs that we have developed together with some examples of
results obtained by letting them run(cid:2)
The applications of the theory (cid:13)the third chapter and numerous examples(cid:14) are
within the (cid:7)eld of mathematical kinetics(cid:1) where one typically has to deal with sys(cid:8)
tems of nonlinear algebraic equations(cid:2) These systems provide a quite broad and
practically signi(cid:7)cant class of examples(cid:2) It should be mentioned that it was this
type of applied problems that originated the team(cid:19)work of the authors(cid:2) To a con(cid:8)
siderable extent our new theoretical results and their computer realizations were
obtained as answers to questions arising in the analysis of chemical kinetic equa(cid:8)
tions(cid:2) As a result of our subsequent work there appeared algorithms and programs
operating on systems with symbolic coe(cid:25)cients of the most general form(cid:2) Our expe(cid:8)
rience with the use of these mathematical and software tools has made us con(cid:7)dent
that the methods of polynomialcomputer algebra willgainfurther development and
be of a wide practical use in diverse (cid:7)elds of science and technology(cid:2)
Since the methods of computer algebra are currently being rapidly developed(cid:1) it
is hardly possible to give a complete account of the subject(cid:1) or even of the branch
related to systems of algebraic equations(cid:2) However(cid:1) the authors do hope that their
experience willprove tobeofsomeusefortheinterested reader(cid:2) Weshallbegrateful
for any comments and requests regarding the subject of this monograph(cid:2)
The material presented in this book was used by the authors in a special courses
of lectures given at the Krasnoyarsk State University(cid:2)
The authors consider it a pleasant duty to express their gratitude toward Lev
Aizenberg for his support during this research and also toward Grigory Yablonskii
for a long and fruitful cooperation(cid:2) On the basis of theoretical results obtained by
Aizenberg we were able to considerably modify the eliminationmethods for systems
of nonlinear algebraic equations(cid:1) and Yablonskii was the one who initiated the use
of new mathematical methods in chemical kinetics(cid:1) and the authors owe to him the
physical and chemical interpretation of a number of joint results(cid:2)
The authors are gratefulto Leonid Bystrov(cid:1) Andrey Levitskii(cid:1) Alexander Semen(cid:8)
ov(cid:1) Vassily Stelletskii(cid:1) Leonid Provorov and Vladimir Topunov for their invaluable
help with the computer realization of this work(cid:1) and also to Alexander Gorban(cid:1)
Vyacheslav Novikov(cid:1) August Tsikh and Alexander Yuzhakov for helpful discussions
in the course of writing the book(cid:2)
We use a double numbering of formulas(cid:1) the (cid:7)rst number denotes the number of
the section(cid:1) and the second indicates the number of the formula within the section(cid:2)
The applications at the end are numbered without reference to particular sections(cid:2)
ix
PREFACE TO THE ENGLISH EDITION
Five years have passed since the publication of our book in Russian(cid:2) In this
period some new results have been obtained(cid:1) and a portion of the material has
become obsolete(cid:2) Therefore(cid:1) apart from the translation work it was also necessary
to revise some sections of the book(cid:2) Let us brie(cid:26)y describe these alterations(cid:2)
In Chapter (cid:9)(cid:1) Section (cid:5)(cid:1) the Theorem (cid:5)(cid:2)(cid:4)and itscorollariesare given inthe more
general situation of projective space and under milder restrictions on the system
of equations(cid:2) In Section (cid:4)(cid:4) we have added some results (cid:13)Theorem (cid:4)(cid:4)(cid:2)(cid:15) and its
corollaries(cid:14) concerning nonlinear and non(cid:8)algebraic systems of equations(cid:2)
In Chapter (cid:11)(cid:1) Section (cid:4)(cid:12)(cid:1) Theorem (cid:4)(cid:12)(cid:2)(cid:6) is proved completely(cid:2) In Section (cid:4)(cid:15) we
give a general survey of the applications of our elimination method to systems of
algebraic and non(cid:8)algebraic equations arising in chemical kinetics(cid:2)
The last chapter has been subject to a thorough revision(cid:2) This is connected with
the appearance of new computer algebra systems(cid:2) In the russian edition we used
the system REDUCE(cid:1) but for the english version we have chosen the Maple system(cid:2)
All software is designed for this system(cid:1) except subsections (cid:4)(cid:16)(cid:2)(cid:11) and (cid:4)(cid:5)(cid:2)(cid:11)(cid:1) where the
previous variant is maintained(cid:2)
We express our thanks to Tatyana Osetrova for software realization of a num(cid:8)
ber of algorithms and to Sergey Znamenskii for consultations regarding the Maple
system(cid:2)
Special gratitude is due to our colleague(cid:1) Professor Mikael Passare(cid:1) who agreed
to undertake the editing of our preliminarytranslation(cid:2) This has helped to substan(cid:8)
tially improve the text(cid:2) The responsibility for any remaining errors(cid:1) mathematical
and linguistic(cid:1) is of course entirely ours(cid:2)
On the (cid:26)oppy disk that comes with the book we have included the software and
the examples described in the fourth chapter(cid:2)
Duringpreparationof the manuscript the (cid:7)rst authorwas partiallysupported by
RFFI(cid:1) grant (cid:5)(cid:11)(cid:8)(cid:4)(cid:9)(cid:8)(cid:17)(cid:17)(cid:15)(cid:5)(cid:10)(cid:1) and the second author was partially supported by RFFI(cid:1)
grant (cid:5)(cid:10)(cid:8)(cid:17)(cid:4)(cid:8)(cid:17)(cid:17)(cid:17)(cid:16)(cid:17)(cid:2)
x
PREFACE BY THE ENGLISH EDITOR
The marvellous discovery(cid:1) that insight into problems on the real line can be
gainedby exploringthe complexplane(cid:1) isas oldas the complexnumbers themselves(cid:2)
Methods from complex analysis are classical tools in countless applications(cid:2) It is
however only fairly recently that ideas from several complex variables have begun to
(cid:7)nd their way into applied mathematics(cid:2) So have pluricomplex methods lately been
introduced in such diverse (cid:7)elds as theoretical physics(cid:1) signal processing(cid:1) medical
tomography and economics(cid:2)
The present monograph(cid:1) written by a well composed troika of one pure math(cid:8)
ematician(cid:1)(cid:1) one computer scientist and one chemist(cid:1) provides not only a thorough
account of the use of multidimensionalresidues in polynomialalgebra(cid:1) together with
computer implementations (cid:3) but also applications of these methods to equilibrium
problems in chemical kinetics(cid:2)
Classical eliminationtheory serves well enough for solvingsmallsystems of poly(cid:8)
nomialequations(cid:1) but for slightlylarger systems it is only of theoreticaluse(cid:2) A more
applicablemethod(cid:1) which has become standard in modern computer algebra(cid:1) is that
of Gr(cid:20)obner bases(cid:2) These allow one to avoid super(cid:26)uous solutions and to bring the
system into triangular form(cid:2)
In this book the central approach is based on logarithmic residues in several
variables(cid:1) the theory of which to a substantial extent has been developed by the
Krasnoyarsk pluricomplex school(cid:1) see for instance the monograph by Aizenberg and
Yuzhakov (cid:13)American Mathematical Society (cid:4)(cid:5)(cid:16)(cid:11)(cid:14) and Tsikh (cid:13)American Mathemat(cid:8)
ical Society (cid:4)(cid:5)(cid:5)(cid:9)(cid:14)(cid:2)
Just as in one variable the method of residues combines elegance with e(cid:25)ciency(cid:2)
To be more precise(cid:1) using a multidimensional analogue of the argument principle
one obtains explicit integral formulas for the number of common roots of the given
system(cid:1) and also for the power sums of these roots(cid:2) This substantially simpli(cid:7)es
the computation of the corresponding resultants(cid:2) In conclusion it is maybe worth
remarking that similar methods occurred already in Poincar(cid:27)e(cid:28)s proof of the Weier(cid:8)
strass preparation theorem(cid:1) and hence can be said to lie at very heart of local
analytic geometry(cid:2)