An algorithm to compute the set of characteristics of a system of polynomial equations over the integers.
type: Item
instance of: scholarly article
title: An algorithm to compute the set of characteristics of a system of polynomial equations over the integers.
zbMATH Open document ID: 1058.13015
DOI: 10.1016/S0747-7171(02)00134-7
author: Rosemary Baines
author: P. Vámos
published in: Journal of Symbolic Computation
publication date: 2003-08-17T00:00:00Z
review text: The question considered in the paper under review is: For a given system of multivariate polynomials with integer coefficients, what are the possible characteristics of fields where this system is solvable? The problem is prompted by and has applications in matroid theory. The authors use the Gröbner bases theory over the integers to provide an algorithm to find this set \(\chi (F)\) for any specific system \(F\). Moreover, they demonstrate that either \(\chi (F)\) is finite and does not contain \(0\), or \(0 \in \chi (F)\) and only finitely many primes are missing from \(\chi (F)\). In the final appendix of the paper it is shown that the result can be proved in a more abstract context (constructible sets of prime ideals in affine algebras over Noetherian domains of Krull dimension 1).
reviewed by: Mihai Cipu
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13-04
Mathematics Subject Classification ID: 11Y05
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 1965168
zbMATH Keywords: Gröbner basis over the integers
zbMATH Keywords: set of characteristics
zbMATH Keywords: matroid
zbMATH Keywords: constructible set of prime ideals
zbMATH Keywords: solvable system of multivariate polynomials
describes a project that uses: Macaulay2
MaRDI profile type: Publication
cites work: Lifting standard bases in filtered structures
cites work: Grobner bases and extension of scalars
cites work: Q4317713
cites work: MATROID REPRESENTATIONS—AN ALGEBRAIC TREATMENT
cites work: Q5626684
cites work: Characteristic Sets of Matroids
cites work: Q5611879
cites work: Q3135082
cites work: Note on Independence Functions
cites work: Q4156448
cites work: Q4152555
cites work: Q4111952
full work available at URL: s0747-7171(02)00134-7
OpenAlex ID: W1995058913
description: scientific article; zbMATH DE number 1965168