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
