Computing representations for radicals of finitely generated differential ideals
type: Item
instance of: scholarly article
title: Computing representations for radicals of finitely generated differential ideals
zbMATH Open document ID: 1185.12003
author: François Boulier
author: D. Lazard
author: François Ollivier
author: M. Petitot
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2009-05-08T00:00:00Z
review text: This article presents an improvement of the Rosenfeld-Gröbner algorithm (details below), which computes a representation for the radical of the differential ideal generated by any system of differential polynomial equations, ordinary or partial. This allows testing membership in the radical ideal and also computing Taylor expansions of solutions of the original system. The core of the algorithm is to express radical differential ideals as intersections of regular differential ideals. The authors have implemented their algorithm as a MAPLE package. The authors improve some results of a previous work [Proceedings of the 1995 international symposium on symbolic and algebraic computation, ISSAC '95, Montreal, Canada, July 10--12, 1995. New York, NY: ACM Press, 158--166 (1995; Zbl 0911.13011)], making the algorithm much more efficient than the previous version. According to them, the Rosenfeld-Gröbner algorithm compares favourably to the similar algorithm by \textit{A. Seidenberg} [Univ. California Publ. Math., n. Ser. 3, 31--66 (1957; Zbl 0083.03302)] and in fact Section 0.3 is devoted to comparing the presented method with other ones in the literature. Section 2 contains an improved proof of Lazard's Lemma, which ``establishes that each regular ideal is radical and that all its prime components have a same parametric set'' including details on performing computations in dimension zero. Section 4 proves the authors' version of Rosenfeld's Lemma which ``gives a sufficient condition so that a system of polynomial differential equations admits a solution if and only if this same system, considered as a purely algebraic system admits a solution'' and auxiliary technical results. Section 5 describes how to represent radical differential ideals as intersections of regular differential ideals. In Section 6 it is shown how to compute canonical representatives for regular differential ideals and the main algorithm is stated as a theorem and proven effectively. Section 7 explains…
reviewed by: David Sevilla
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13P99
zbMATH DE Number: 5552161
zbMATH Keywords: Rosenfeld-Gröbner algorithm
zbMATH Keywords: systems of polynomial differential equations
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: s00200-009-0091-7
OpenAlex ID: W2054452549
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DOI: 10.1007/S00200-009-0091-7
description: scientific article; zbMATH DE number 5552161