In recent years, the discovery of new algorithms for dealing with polynomial equations, coupled with their implementation on inexpensive yet fast computers, has sparked a minor revolution in the study and practice of algebraic geometry. One of the aims of this text is to illustrate the various uses of algebraic geometry and to highlight the more recent applications of Groebner bases and resultants. In order to do this, an introduction to some advanced algebraic objects and techniques is provided.

Recensione:

"Das vorliegende Buch ist exzellent geschrieben. Es enth�lt neben zahlreichen theoretischen auch viele praktische Aufgaben mit Computeralgebrasystemen ( vor allem Macaulay, Maple und Singular). Zusammen mit Cox, Little, O'Shea: Ideals, Varieties, and Algorithms ist es sehr gut geeignet als erste Einf�hrung in die angesprochenen Themenkreise. Dabei k�nnen sehr viele Kapitel des Buchs im Wesentlichen unabh�ngig von den anderen Kapiteln gelesen werden."
DMV Jahresbericht, Bd. 103, Heft 1, 2001

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