A modular method to compute the rational univariate representation of zero-dimensional ideals
type: Item
instance of: scholarly article
title: A modular method to compute the rational univariate representation of zero-dimensional ideals
zbMATH Open document ID: 0945.13010
author: Masayuki Noro
author: Kazuhiro Yokoyama
published in: Journal of Symbolic Computation
publication date: 2000-10-04T00:00:00Z
review text: Given an ideal \(I \subseteq K[x _1,\ldots,x _n]\), it is a difficult task to determine the set of zeros of \(I\), even if \(I\) is known to be zero-dimensional. An important tool for this is the use of Gröbner bases, which by the so-called shape lemma often have a form suitable for finding zeros. The authors propose a new method to find a rational univariate representation (RUR) for \(I\), as defined by \textit{F. Rouillier} [Appl. Algebra Eng. Commun. Comput. 9, No.~5, 433-461 (1999; Zbl 0932.12008)]. This is another ideal basis which has the form \[ \{f(u),g _1(u) x _1 - h _1(u),\ldots,g _n(u) x _n - h _n(u)\} \] with \(u \in K[x _1,\ldots,x _n]\) a ``separating element'' and \(f,g _i,h _i\) univariate polynomials. The advantage over a Gröbner basis is that an RUR often has smaller coefficients. The new method applies to the case where the ground field \(K\) is \(\mathbb Q\) and uses reduction modulo a prime and Hensel lifting.
reviewed by: Gregor Kemper
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1369547
zbMATH Keywords: zeros of ideals
zbMATH Keywords: modular methods
zbMATH Keywords: shape lemma
zbMATH Keywords: rational univariate representation
zbMATH Keywords: Gröbner basis
zbMATH Keywords: Hensel lifting
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1369547