Explicit polynomial bounds on prime ideals in polynomial rings over fields

William Simmons and Henry Towsner

Vol. 306 (2020), No. 2, 721–754

DOI: 10.2140/pjm.2020.306.721

Abstract

Consider an ideal $I\subseteq k\left[x_1,\dots ,x_n\right]$ of a polynomial ring over a field with the property that for some $b$, if $fg\in I$ for $f,g$ of degree $\le b$, 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  $k\left[x_1,\dots ,x_n\right]$.

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