Table of Contents of Ideals, Varieties, and Algorithms
Fourth Edition, 2015

Chapter 1: Geometry, Algebra, and Algorithms

  1. Polynomials and Affine Space
  2. Affine Varieties
  3. Parametrizations of Affine Varieties
  4. Ideals
  5. Polynomials of One Variable

Chapter 2: Gröbner Bases

  1. Introduction
  2. Orderings on the Monomials in k[x1,...,xn]
  3. A Division Algorithm in k[x1,...,xn]
  4. Monomial Ideals and Dickson's Lemma
  5. The Hilbert Basis Theorem and Gröbner Bases
  6. Properties of Gröbner Bases
  7. Buchberger's Algorithm
  8. First Applications of Gröbner Bases
  9. Refinements of the Buchberger Criterion (New in the Fourth Edition)
  10. Improvements on Buchberger's Algorithm

Chapter 3: Elimination Theory

  1. The Elimination and Extension Theorems
  2. The Geometry of Elimination
  3. Implicitization
  4. Singular Points and Envelopes
  5. Gröbner Bases and the Extension Theorem (New in the Fourth Edition)
  6. Resultants and the Extension Theorem

Chapter 4: The Algebra-Geometry Dictionary

  1. Hilbert's Nullstellensatz
  2. Radical Ideals and the Ideal-Variety Correspondence
  3. Sums, Products, and Intersections of Ideals
  4. Zariski Closures, Ideal Quotients, and Saturations
  5. Irreducible Varieties and Prime Ideals
  6. Decomposition of a Variety into Irreducibles
  7. Proof of the Closure Theorem (New in the Fourth Edition)
  8. Primary Decomposition of Ideals
  9. Summary

Chapter 5: Polynomial and Rational Functions on a Variety

  1. Polynomial Mappings
  2. Quotients of Polynomials Rings
  3. Algorithmic Computations in k[x1,...,xn]/I
  4. The Coordinate Ring of an Affine Variety
  5. Rational Functions on a Variety
  6. Relative Finiteness and Noether Normalization (New in the Fourth Edition)

Chapter 6: Robotics and Automatic Geometric Theorem Proving

  1. Geometric Description of Robots
  2. The Forward Kinematics Problem
  3. The Inverse Kinematic Problem and Motion Planning
  4. Automatic Geometric Theorem Proving
  5. Wu's Method

Chapter 7: Invariant Theory of Finite Groups

  1. Symmetric Polynomials
  2. Finite Matrix Groups and Rings of Invariants
  3. Generators for the Ring of Invariants
  4. Relations among Generators and the Geometry of Orbits

Chapter 8: Projective Algebraic Geometry

  1. The Projective Plane
  2. Projective Space and Projective Varieties
  3. The Projective Algebra-Geometry Dictionary
  4. The Projective Closure of an Affine Variety
  5. Projective Elimination Theory
  6. The Geometry of Quadric Hypersurfaces
  7. Bezout's Theorem

Chapter 9: The Dimension of a Variety

  1. The Variety of a Monomial Ideal
  2. The Complement of a Monomial Ideal
  3. The Hilbert Function and the Dimension of a Variety
  4. Elementary Properties of Dimension
  5. Dimension and Algebraic Independence
  6. Dimension and Nonsingularity
  7. The Tangent Cone

Chapter 10: Additional Gröbner Basis Algorithms (New in the Fourth Edition)

  1. Preliminaries
  2. Hilbert Driven Buchberger Algorithm
  3. The F4 Algorithm
  4. Signature-based Algorithms and F5

Appendix A: Some Concepts from Algebra

  1. Fields and Rings
  2. Unique Factorization
  3. Groups
  4. Determinants

Appendix B: Pseudocode

  1. Inputs, Outputs, Variables and Constants
  2. Assignment Statements
  3. Looping Structures
  4. Branching Structures
  5. Output Statements

Appendix C: Computer Algebra Systems

  1. General Purpose Systems: Maple, Mathematica, Sage
  2. Special Purpose Programs: CoCoA, Macaulay2, Singular
  3. Other Systems and Packages

Appendix D: Independent Projects

  1. General Comments
  2. Suggested Projects

References

Index

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