Integrally Hilbertian rings and the polynomial Schinzel hypothesis

TL;DR

This paper introduces integrally Hilbertian rings, extending Hilbert's specialization property to rings, and applies this to a polynomial version of the Schinzel Hypothesis, broadening its scope.

Contribution

It develops a criterion for integral Hilbertianity in rings, including rings of integers of number fields, and applies it to generalize the Schinzel Hypothesis.

Findings

Established a criterion for integral Hilbertianity in Krull domains.

Proved a polynomial variant of the Schinzel Hypothesis over integrally Hilbertian rings.

Generalized previous results on prime values of polynomials to a broader algebraic setting.

Abstract

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the notion of {integrally Hilbertian rings}, where specialization takes place inside a ring and irreducibility is required over the ring. A core part shows how new obstacles to irreducibility such as coefficient divisors or fixed divisors can be dealt with over Krull domains, a large class of rings including UFDs, Dedekind domains, etc. As a result, we obtain a general criterion for integral hilbertianity, along with many examples, \hbox{e.g.} all rings of integers of number fields. Polynomial rings over arbitrary domains are other examples. As an application, we prove a polynomial variant of the Schinzel Hypothesis on prime values of polynomials with integer coefficients: if $\mathcal{Z}$ is an integrally Hilbertian ring, the hypothesis becomes a true statement if the ring of integers ${\mathbb Z}$ is replaced by the polynomial ring $\mathcal{Z}[U]$ and ``prime'' by ``irreducible''. This result generalizes previous works and fits in a unified framework for Schinzel-type phenomena that we introduce. We further obtain an additional conclusion that has some noteworthy consequences for the classical Schinzel Hypothesis itself.