Polynomial bounds in polynomial rings over fields
type: Item
instance of: scholarly article
title: Polynomial bounds in polynomial rings over fields
zbMATH Open document ID: 0715.13016
DOI: 10.1016/0021-8693(89)90299-8
author: Karsten Schmidt-Göttsch
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \(K\) be a field and let \(I\) be an ideal in the polynomial ring \(K[X] = K[X_1,\ldots,X_n]\). Suppose that \(I\) is generated by polynomials of (total) degree at most \(d\). The author uses non-standard set theory to give the following primality criterion. There is an integer \(\beta =\beta(n)\), independent of the field \(K\), so that \(I\) is prime if and only if the following holds: \(1\not\in I\), and for all \(f,g\in K[X]\) of degree \(\leq d^{\beta}\), if \(fg\in I\) then either \(f\in I\) or \(g\in I\). A principal result proved en route is that any system of linear equations over \(K[X]\) with coefficients of bounded degree \(d\) must have a solution (if any) of degree at most \(d^{\alpha}\), where \(\alpha\) depends only on \(n\).
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 12E12
Mathematics Subject Classification ID: 12L15
zbMATH DE Number: 4177237
zbMATH Keywords: ideal in the polynomial ring
zbMATH Keywords: non-standard set theory
zbMATH Keywords: primality criterion
zbMATH Keywords: system of linear equations
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90299-8
OpenAlex ID: W2075776007
cites work: Bounds in the theory of polynomial rings over fields. A nonstandard approach
cites work: Q5588717
cites work: Constructions in Algebra
cites work: Q5511421
cites work: Nonstandard arithmetic of function fields over H-convex subfields of *Q.
cites work: Bounds for the degrees in the Nullstellensatz
cites work: The complexity of the word problems for commutative semigroups and polynomial ideals
Recommended article: Bounds in the theory of polynomial rings over fields. A nonstandard approach
Recommended article: Bounds and definability in polynomial rings
Recommended article: Polynomial bounds for rings of invariants
Recommended article: Explicit polynomial bounds on prime ideals in polynomial rings over fields
Recommended article: Bounds in polynomial rings over Artinian local rings
Recommended article: Q4049191
Recommended article: Some Boundedness Conditions for Rings of Integer-Valued Polynomials
Recommended article: On polynomial rings
Recommended article: Q3710618
Recommended article: Q4790086
description: scientific article; zbMATH DE number 4177237