多项式方程组的求解是数学中的经典问题。今天，多项式模型无处不在，并在科学中广泛使用，如机器人技术、编码理论、优化、数学生物学、计算机视觉、博弈论、统计学及许多其他领域。本书提供了跨越数学学科的桥梁，揭示了多项式方程组的许多方面。它涵盖了广泛的数学技巧和算法，包括符号计算和数值计算。
多项式方程组的解集是代数变量——代数几何的基本对象。代数变量的算法研究是计算代数几何的核心主题。几何计算软件的最新发展令人兴奋，已经彻底改变了这个领域。以前棘手的问题已易于处理，这为实验和猜想提供了沃土。
本书的前半部分简要介绍了计算代数几何的最新技术，即代数簇的算法研究；后半部分从各种新颖和意想不到的角度探讨了多项式方程，介绍了学科间的联系，讨论了当前研究的重点，并概述了未来可能的算法。
整本书中有许多动手实例和练习，包括Mapie，MATLAB，Macaulay 2，Singular，PHCpack，SOSTools和CoCoA的简短但完整的会话。这些例子对没有代数几何或交换代数背景的读者特别有用。几分钟之内，读者就能学会如何输入多项式方程，并在计算机屏幕上看到一些有意义的结果。
读者需要具备基本的抽象和计算代数知识。本书适合作为计算代数方向的研究生课程教材。

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 Multivariate 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.Positive 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