The Gröbner cover
type: Item
instance of: scholarly article
title: The Gröbner cover
zbMATH Open document ID: 1412.13001
DOI: 10.1007/978-3-030-03904-2
author: Antonio Montes
published in: Algorithms and Computation in Mathematics
publication date: 2019-01-10T00:00:00Z
review text: Gröbner bases turned out to be a powerful tool to study polynomial systems. The aim of the book is to study polynomial systems with parameters. \textit{V. Weispfenning} [J. Symb. Comput. 14, No. 1, 1--29 (1992; Zbl 0784.13013)] introduced comprehensive Gröbner bases, a tool which can handle polynomial systems with parameters. Let $K$ be a field and $\overline{K}$ its algebraic closure. Let $u=(u_1, \ldots, u_m)$ a set of parameters and $x=(x_1, \ldots, x_n)$ a set of variables. Let $I\subseteq K[u][x]$ be an ideal and for $a\in \overline{K}^m$ let $I_a$ be the image of $I$ in $\overline{K}[x]$ substituting $u=a$. A comprehensive Gröbner basis is a finite subset $G\subset I$ that specializes for every $a\in \overline{K}^m$ to a Gröbner basis of $I_a$. The Gröbner cover (introduced by [\textit{A. Montes} and \textit{M. Wibmer}, J. Symb. Comput. 45, No. 12, 1391--1425 (2010; Zbl 1207.13018)]) is the analog to reduced Gröbner bases of an ideal for parametric ideals. There is a canonical partition $\overline{K}^m=\overset{i}{\underset{i=1}{\bigcup}}S_i$ into disjoint locally closed subsets. For each $S_i$ there exists a generalized Gröbner basis, represented by a set of polynomial functions $B_i$ that specialize for $a\in S_i$ to the reduced Gröbner basis of $I_a$. If the ideal $I$ is homogeneous with respect to $x$ the $S_i$ are characterized by fixing the leading ideal of $I_a$. The Gröbner cover is the set $\{(S_1, B_1), \ldots, (S_r, B_r)\}$. An algorithm to compute the Gröbner cover was implemented by the author of the book 2010 as a library in the computer algebra system \textsc{Singular}. The book contains two parts, a theoretical part and a part with applications. The first chapter gives some basic results about Gröbner bases. Chapter 2, 3 and 4 contain basic tools needed for the construction of the Gröbner cover. In chapter 2 constructible sets are explained and algorithms to compute them. Chapter 3 introduces comprehensive Gröbner systems and bases and discusse…
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 6999636
zbMATH Keywords: Gröbner basis
zbMATH Keywords: comprehensive Gröbner basis
zbMATH Keywords: polynomial systems with parameters
describes a project that uses: SINGULAR
describes a project that uses: GeoGebra
describes a project that uses: RegularChains
describes a project that uses: REDLOG
describes a project that uses: grobcov.lib
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-03904-2
OpenAlex ID: W2919779544
description: scientific article; zbMATH DE number 6999636