Decker W. A First Course in Computational Algebraic Geometry 2013

Textbook in PDF format

Most of mathematics is concerned at some level with setting up and solving various types of equations. Algebraic geometry is the mathematical discipline which handles solution sets of systems of polynomial equations. These are called algebraic sets. Making use of a correspondence which relates algebraic sets to ideals in polynomial rings, problems concerning the geometry of algebraic sets can be translated into algebra. As a consequence, algebraic geometers have developed a multitude of often highly abstract techniques for the qualitative and quantitative study of algebraic sets, without, in the first instance, considering the equations. Modern computer algebra algorithms, on the other hand, allow us to manipulate the equations and, thus, to study explicit examples. In this way, algebraic geometry becomes accessible to experiments. The experimental method, which has proven to be highly successful in number theory, is now also added to the toolbox of the algebraic geometer. In these notes, we discuss some of the basic operations in geometry and describe their counterparts in algebra. We explain how the operations can be carried through using computer algebra methods, and give a number of explicit examples, worked out with the computer algebra system Singular. In this way, our book may serve as a first introduction to Singular, guiding the reader to performing his own experiments