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