Solving Systems of Polynomial Equations

CBMS - Conference Board of the Mathemetical Sciences

Bernd Sturmfels

152 pages, parution le 08/04/2003

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Résumé

A classic problem in mathematics is solving systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely used across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, and numerous other areas.

This book furnishes a bridge across mathematical disciplines and exposes many facets of systems of polynomial equations. It covers a wide spectrum of mathematical techniques and algorithms, both symbolic and numerical.

The set of solutions to a system of polynomial equations is an algebraic variety--the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in computer software for geometric calculations have revolutionized the field. Formerly inaccessible problems are now tractable, providing fertile ground for experimentation and conjecture.

The first half of the book gives a snapshot of the state of the art of the topic. Familiar themes are covered in the first five chapters, including polynomials in one variable, Gröbner bases of zero-dimensional ideals, Newton polytopes and Bernstein's Theorem, multidimensional resultants, and primary decomposition.

The second half of the book explores polynomial equations from a variety of novel and unexpected angles. It introduces interdisciplinary connections, discusses highlights of current research, and outlines possible future algorithms. Topics include computation of Nash equilibria in game theory, semidefinite programming and the real Nullstellensatz, the algebraic geometry of statistical models, the piecewise-linear geometry of valuations and amoebas, and the Ehrenpreis-Palamodov theorem on linear partial differential equations with constant coefficients.

Throughout the text, there are many hands-on examples and exercises, including short but complete sessions in Maple®, MATLAB®, Macaulay 2, Singular, PHCpack, CoCoA, and SOSTools. These examples will be particularly useful for readers with no background in algebraic geometry or commutative algebra. Within minutes, readers can learn how to type in polynomial equations and actually see some meaningful results on their computer screens.

Contents

Polynomials in one variable
Groebner bases of zero-dimensional ideals
Bernstein's theorem and fewnomials
Resultants
Primary decomposition
Polynomial systems in economics
Sums of squares
Polynomial systems in statistics
Tropical algebraic geometry
Linear partial differential equations with constant coefficients
Bibliography
Index

L'auteur - Bernd Sturmfels

Bernd Sturmfels received doctoral degrees in 1987 from the University of Washington, Seattle and TU Darmstadt, Germany. After two postdoc years at the IMA in Minneapolis and RISC-Linz in Austria, he taught at Cornell University before joining UC Berkeley in 1995, where he is now Professor of Mathematics and Computer Science. A leading experimentalist among mathematicians, Sturmfels has authored seven books and over 130 research articles in the areas of combinatorics, algebraic geometry, symbolic computation, and their applications, and he has mentored 16 doctoral students.

Caractéristiques techniques

	PAPIER
Éditeur(s)	American Mathematical Society (AMS)
Auteur(s)	Bernd Sturmfels
Parution	08/04/2003
Nb. de pages	152
Format	17,5 x 25
Couverture	Broché
Poids	285g
Intérieur	Noir et Blanc
EAN13	9780821832516
ISBN13	978-0-8218-3251-6