Elimination methods
type: Item
instance of: scholarly article
title: Elimination methods
zbMATH Open document ID: 0964.13014
author: Dongming Wang
published in: Texts \& Monographs in Symbolic Computation
publication date: 1999-12-08T00:00:00Z
review text: The task of solving a system of polynomial equations, or more generally of extracting information from it, arises in a variety of situations, ranging from fields of pure mathematics like algebraic geometry to applied mathematics and to technical applications like robotics. Elimination is a technique for addressing these problems which has a long history. Born from the geometric idea of considering the projection of a given object or, equivalently, the intersection of an ideal in a polynomial ring with a subring involving fewer variables, it has been an important constructive tool for a long time. After loosing importance when more elegant abstract arguments appeared in commutative ring theory, a renaissance of elimination methods followed the progress of computer technology, since it is nowadays feasible to perform the rather complex computations algorithmically on a computer. The book ``Elimination methods'' by D.~Wang under review presents different elimination algorithms that are presently in use. After briefly introducing basic concepts and properties of multivariate polynomials, the author focuses on algorithms which decompose arbitrary polynomial systems into triangular systems, that is, into systems of polynomials \(T_1,\dots,T_r\) in which the polynomial \(T_i\) only involves the first \(p_i\) variables of the base ring \((p_1 < \ldots < p_r)\). Before proceeding to some applications of elimination in the last chapters, other better-known algorithms based on Gröbner bases and on resultant techniques are explained, but not in as much detail as in the discussion of triangular systems. The discussion of each method in this book also contains a formal description of the algorithm, which allows a rather direct implementation, and some examples illustrating it; complexity questions and practical implementation are not covered in this book.
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13-01
Mathematics Subject Classification ID: 68-01
Mathematics Subject Classification ID: 14Q99
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1376211
zbMATH Keywords: elimination
zbMATH Keywords: triangular sets
zbMATH Keywords: resultants
zbMATH Keywords: Gröbner bases
zbMATH Keywords: polynomial equations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1376211