Explicit polynomial bounds on prime ideals in polynomial rings over fields

TL;DR

This paper provides explicit polynomial bounds on the degree needed to determine when an ideal in a polynomial ring over a field is prime or maximal, improving upon previous non-constructive results.

Contribution

It offers explicit polynomial bounds on the degree for primality and maximality detection in polynomial rings, advancing prior model-theoretic approaches.

Findings

Derived explicit polynomial bounds for prime ideals.

Extended bounds to maximal ideals detection.

Improved understanding of degree conditions in polynomial rings.

Abstract

Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime. Schmidt-G\"ottsch proved that "sufficiently large" can be taken to be a polynomial in the degree of generators of $I$ (with the degree of this polynomial depending on $n$). However Schmidt-G\"ottsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on $b$, polynomial in the degree of the generators of $I$. We also give a similar bound for detecting maximal ideals in $k[x_1,\ldots,x_n]$.