Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence
type: Item
instance of: scholarly article
title: Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence
zbMATH Open document ID: 0681.68052
DOI: 10.1016/S0747-7171(88)80047-6
author: David Shannon
author: Moss Sweedler
published in: Journal of Symbolic Computation
publication date: 1988-01-01T00:00:00Z
review text: Let \(X_ 1,...,X_ n\), \(T_ 1,...,T_ m\) be indeterminates over the field k and let \(g,f_ 1,...,f_ m\in k[X_ 1,...,X_ n]\). An algorithm involving Gröbner bases is given which will determine if \(g\in k[f_ 1,...,f_ m]\), and if so will produce \(P\in k[T_ 1,...,T_ m]\) such that \(g=P(f_ 1,...,f_ m)\). Thus for a k-algebra homomorphism \(\omega\) : \(B\to k[X_ 1,...,X_ n]\), where B is a finitely generated k-algebra, the algorithm will determine if \(\omega\) is onto, and if so it will produce a right inverse of \(\omega\). To explain the algorithm further choose a term ordering on the monomials of \(k[X_ 1,...,X_ n\), \(T_ 1,...,T_ m]\) such that each \(X_ i\) is larger than any monomial in \(k[T_ 1,...,T_ m]\). Let J be the ideal of \(k[X_ 1,...,X_ n\), \(T_ 1,...,T_ m]\) generated by \(\{f_ 1-T_ 1,...,f_ m-X_ m\}\), and let G be a Gröbner basis for J. Then for \(g\in k[X_ 1,...,X_ n]\) we may reduce g over G until reduction terminates, say with \(h\in k[X_ 1,...,X_ n\), \(T_ 1,...,T_ m]\). Then \(g\in k[X_ 1,....,X_ n]\) if and only if \(h\in k[T_ 1,...,T_ m]\) and in this case \(g=h(f_ 1,...,f_ m)\). Some related results given are tests for when \(k[f_ 1,...,f_ m]=k[X_ 1,...,X_ n]\) and when \(k(f_ 1,...,f_ m)=k(X_ 1,...,X_ n)\).
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 4115957
zbMATH Keywords: Gröbner bases
reviewed by: David E. Rush
MaRDI profile type: Publication
Wikidata QID: Q56287868
cites work: Comments on the translation of my PhD thesis: ``An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal''
cites work: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems
cites work: Q3208084
cites work: Q3714165
description: scientific article; zbMATH DE number 4115957