Explicit polynomial bounds on prime ideals in polynomial rings over fields

Consider an ideal $I \subseteq k [x_{1}, \dots, x_{n}]$ of a polynomial ring over a field with the property that for some $b$ , if $f g \in I$ for $f, g$ of degree $\leq b$ , then $f \in I$ or $g \in I$ . It is known that if $b$ is sufficiently large, then $I$ is prime. We construct an explicit bound on $b$ , polynomial in the degree of the generators of $I$ (the existence of such a bound was established by Schmidt-Göttsch in 1989). We also give a similar bound for detecting maximal ideals in $k [x_{1}, \dots, x_{n}]$ .

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Keywords

uniform bounds, prime ideals, maximal ideals, proof mining, Gröbner bases

Mathematical Subject Classification 2010

Primary: 12Y05

Secondary: 12L10

Milestones

Received: 7 June 2019

Revised: 28 January 2020

Accepted: 30 January 2020

Published: 13 July 2020

Authors

William Simmons
Department of Mathematics and Computer Science Hobart and William Smith Colleges Geneva, NY United States

Henry Towsner
Department of Mathematics University of Pennsylvania Philadelphia, PA United States

Vol. 306, No. 2, 2020

William Simmons and Henry Towsner

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors