Computing bases of complete intersection rings in Noether position
type: Item
instance of: scholarly article
title: Computing bases of complete intersection rings in Noether position
zbMATH Open document ID: 1029.13016
DOI: 10.1016/S0022-4049(00)00135-3
author: Marcela Almeida
author: Manuela Blaum
author: Lisi D'Alfonso
author: Pablo Solernó
published in: Journal of Pure and Applied Algebra
publication date: 2004-02-03T00:00:00Z
reviewed by: Alessandro Logar
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W10
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13-04
zbMATH DE Number: 1658968
zbMATH Keywords: Quillen-Suslin theorem
zbMATH Keywords: Noether position
zbMATH Keywords: computing bases of free modules
zbMATH Keywords: computing bases for complete intersection rings
MaRDI profile type: Publication
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review text: Assume that \(k\) is an effective infinite perfect field. The aim of the paper is to compute bases of certain free modules over the ring of polynomials \(k[x_1,\dots, x_n]\) and to give a uniform setting to the solution of the problems (which, in some cases, can already be found in the literature). Among the results of the paper we have: NEWLINENEWLINENEWLINELet \(F\) be an \(M \times M\) matrix with polynomial entries such that \(F^2=F\) (i.e. \(F\) is a linear projection), assume that the degrees of the entries of \(F\) are bounded by an integer \(D\) and are given by a straight-line program of size \(L\). Then there exists a well parallelizable algorithm which computes (in sequential time \((nL)^{O(1)}(MD)^{O(n)})\) two subsets of \(k[x_1,\dots,x_n]^M\): \(\{v_1,\dots,v_s\}\) and \(\{v_{s+1}, \dots,v_M\}\) such that:NEWLINENEWLINENEWLINE(1) \(\{v_1,\dots,v_M\}\) is a basis of \(k[x_1,\dots,x_n]^M\); NEWLINENEWLINE(2) \(\{v_1,\dots,v_s\}\) is a basis of \(\text{Im}(F)\) and \(\{v_{s+1},\dots,v_M\}\) is a basis of \(\text{Ker}(F)\). Moreover the entries of \(v_1,\dots,v_M\) are bounded in degree by \((MD)^{O(n)}\).NEWLINENEWLINENEWLINEFurthermore, given a finitely generated module \(P\) over the polynomial ring, the authors show that there exists a single exponential algorithm which decides if \(P\) is free and in the affirmative case, computes a free basis of it.NEWLINENEWLINENEWLINEThe above result can be used to compute bases for complete intersection rings in Noether position. More precisely: NEWLINENEWLINENEWLINELet \(f_1,\dots, f_{n-r} \in k[x_1,\dots,x_n]\) be a regular sequence of polynomials of degrees bounded by an integer \(d\) and given by a straight-line program of size \(l\), such that the canonical morphism \( k[x_1,\dots,x_r] \rightarrow S:=k[x_1,\dots,x_n]/( f_1,\dots,f_{n-r})\) is injective and integral, then there exists a single exponential algorithm which computes a basis of \(S\) as a free \( k[x_1,\dots,x_r]\)-module. The proposed algorithm i…
description: scientific article; zbMATH DE number 1658968