Explicit polynomial bounds on prime ideals in polynomial rings over fields

Abstract

Suppose I is an ideal of a polynomial ring over a field, I ⊆ k[x1, . . . , xn], and whenever f g ∈ I for f, g of degree ≤ b, then either f ∈ I or g ∈ I. When b is sufficiently large, it turns out that I is prime. Schmidt-Göttsch proved [24] 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-Göttsch used modeltheoretic 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 [x1, . . . , xn].