●Preface
Chapter 1.Polynomials in One Variable
1.1.The Fundamental Theorem of Algebra
1.2.Numerical Root Finding
1.3.Real Roots
1.4.Puiseux Series
1.5.Hypergeometric Series
1.6.Exercises
Chapter 2.GrSbner Bases of Zero-Dimensional Ideals
2.1.Computing Standard Monomials and the Radical
2.2.Localizing and Removing Known Zeros
2.3.Companion Matrices
2.4.The Trace Form
2.5.Solving Polynomial Equations in Singular
2.6.Exercises
Chapter 3.Bernstein's Theorem and Fewnomials
3.1.From Bzout's Theorem to Bernstein's Theorem
3.2.Zero-dimensional Binomial Systems
3.3.Introducing a Toric Deformation
3.4.Mixed Subdivisions of Newton Polytopes
3.5.Khovanskii's Theorem on Fewnomials
3.6.Exercises
Chapter 4.Resultants
4.1.The Univariate Resultant
4.2.The Classical ltivariate Resultant
4.3.The Sparse Resultant
4.4.The Unmixed Sparse Resultant
4.5.The Resultant of Four Trilinear Equations
4.6.Exercises
Chapter 5.Primary Decomposition
5.1.Prime Ideals, Radical Ideals and Primary Ideals
5.2.How to Decompose a Polynomial System
5.3.Adjacent Minors
5.4.Permanental Ideals
5.5.Exercises
Chapter 6.Polynomial Systems in Economics
6.1.Three-Person Games with Two Pure Strategies
6.2.Two Numerical Examples Involving Square Roots
6.3.Equations Defining Nash Equilibria
6.4.The Mixed Volume of a Product of Simplices
6.5.Computing Nash Equilibria with PHCpack
6.6.Exercises
Chapter 7.Sums of Squares
7.1. itive Semidefinite Matrices
7.2.Zero-dimensional Ideals and SOStools
7.3.Global Optimization
7.4.The Real Nullstellensatz
7.5.Symmetric Matrices with Double Eigenvalues
7.6.Exercises
Chapter 8.Polynomial Systems in Statistics
8.1.Conditional Independence
8.2.Graphical Models
8.3.Random Walks on the Integer Lattice
8.4.Maximum Likelihood Equations
8.5.Exercises
Chapter 9.Tropical Algebraic Geometry
9.1.Tropical Geometry in the Plane
9.2.Amoebas and their Tentacles
9.3.The Bergman Complex of a Linear Space
9.4.The Tropical Variety of an Ideal
9.5.Exercises
Chapter 10.Linear Partial Differential Equations with Constant Coefficients
10.1.Why Differential Equations?
10.2.Zero-dimensional Ideals
10.3.Computing Polynomial Solutions
10.4.How to Solve Monomial Equations
10.5.The Ehrenpreis-Palamodov Theorem
10.6.Noetherian Operators
10.7.Exercises
Bibliography
Index

多項式方程組的求解是數學中的經典問題。今天，多項式模型無處不在，並在科學中廣泛使用，如機器人技術、編碼理論、優化、數學生物學、計算機視覺、博弈論、統計學及許多其他領域。本書提供了跨越數學學科的橋梁，揭示了多項式方程組的許多方面。它涵蓋了廣泛的數學技巧和算法，包括符號計算和數值計算。
多項式方程組的解集是代數變量――代數幾何的基本對像。代數變量的算法研究是計算代數幾何的核心主題。幾何計算軟件的近期新發展令人興奮，已經改變了這個領域。以前棘手的問題已易於處理，這為實驗和猜想提供了沃土。
本書的前半部分簡要介紹了計算代數幾何的近期新技術，即代數簇的算法研究；後半部分從各種新穎和意想不到的角度探討了多項式方程，介紹了學科間的聯繫，討論了當前研究的重點，並概述了未來可能的算法。