#### Vol. 306, No. 2, 2020

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Explicit polynomial bounds on prime ideals in polynomial rings over fields

### William Simmons and Henry Towsner

Vol. 306 (2020), No. 2, 721–754
##### 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
Primary: 12Y05
Secondary: 12L10