Ideals, Varieties, and Algorithms

Written at an undergraduate level, this textbook emphasizes applications and the computational and algorithmic aspects of algebraic geometry and commutative algebra. It covers such topics as the Hilbert Basis Theorem, invariant theory, protective geometry and dimensional theory.

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This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960s. 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 led to some interesting applications, for example in robotics and in geometric theorem proving. This book is an introduction to algebraic geometry and commutative algebra aimed primarily at undergraduates. Emphasizing applications and the computational and algorithmic aspects of the subject, the text has less abstract flavour than standard treatments. With few prerequisites, it should also a useful introduction to the subject for computer scientists.