Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The book bases its discussion of algorithms on a generalisation of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing this new edition, the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem.

Inhalt

1. Geometry, Algebra, and Algorithms.- 2. Groebner Bases.- 3. Elimination Theory.- 4. The Algebra-Geometry Dictionary.- 5. Polynomial and Rational Functions on a Variety.- 6. Robotics and Automatic Geometric Theorem Proving.- 7. Invariant Theory of Finite Groups.- 8. Projective Algebraic Geometry.- 9. The Dimension of a Variety.- Appendix A. Some Concepts from Algebra.- Fields and Rings.- Groups.- Determinants.- Appendix B. Pseudocode.- Inputs, Outputs, Variables, and Constants.- Assignment Statements.- Looping Structures.- Branching Structures.- Appendix C. Computer Algebra Systems.- AXIOM.- Maple.- Mathematica.- REDUCE.- Other Systems.- Appendix D. Independent Projects.- General Comments.- Suggested Projects.- References.