**CBMS Regional Conference Series in Mathematics**

Volume: 97;
2002;
152 pp;
Softcover

MSC: Primary 13; 14; 65;
Secondary 12; 35; 52; 62; 68; 90; 91

Print ISBN: 978-0-8218-3251-6

Product Code: CBMS/97

List Price: $40.00

Individual Price: $32.00

**Electronic ISBN: 978-1-4704-2457-2
Product Code: CBMS/97.E**

List Price: $40.00

Individual Price: $32.00

# Solving Systems of Polynomial Equations

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*Bernd Sturmfels*

A co-publication of the AMS and CBMS

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 software. 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.

Prerequisites include basic abstract and computational algebra. The book is
designed as a text for a graduate course in computational algebra.

#### Readership

Graduate students and research mathematicians interested in computational algebra and its applications.

#### Reviews & Endorsements

Methods for solving systems of polynomial equations in the tropical semiring promise to have wide-ranging applications and have not been treated in monograph before. The book is written in an lively style with many examples, comments, computer algebra sessions and a generous dose of tempting exercises. It can be read with a reasonably good knowledge of basic algebra and is well suited for a lecture course on the graduate level. For the researcher who wants to get an accessible introduction to current techniques in polynomial system solving it can be highly recommended.

-- Zentralblatt Math

#### Table of Contents

# Table of Contents

## Solving Systems of Polynomial Equations

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface vii8 free
- Chapter 1. Polynomials in One Variable 110 free
- Chapter 2. Gröbner Bases of Zero-Dimensional Ideals 1322
- Chapter 3. Bernstein's Theorem and Fewnomials 2938
- Chapter 4. Resultants 4352
- Chapter 5. Primary Decomposition 5968
- Chapter 6. Polynomial Systems in Economics 7180
- Chapter 7. Sums of Squares 8796
- Chapter 8. Polynomial Systems in Statistics 101110
- Chapter 9. Tropical Algebraic Geometry 119128
- Chapter 10. Linear Partial Differential Equations with Constant Coefficients 133142
- Bibliography 147156
- Index 151160
- Back Cover Back Cover1162