The relative Schinzel hypothesis

TL;DR

This paper introduces a relative version of the Schinzel Hypothesis, extending its scope to various integral domains and providing new results on prime and irreducible values of polynomials.

Contribution

It formulates a relative Schinzel Hypothesis and proves it for several classes of integral domains, also deriving an integral version of the Hilbert Irreducibility Theorem.

Findings

Proved the relative Schinzel Hypothesis for PIDs, UFDs with infinite fields, and polynomial rings over UFDs.

Established conditions under which polynomials assume relatively prime values at some integer.

Developed an integral version of the Hilbert Irreducibility Theorem with irreducibility over the ring.

Abstract

The Schinzel Hypothesis is a conjecture about irreducible polynomials in one variable over the integers: under some standard condition, they should assume infinitely many prime values at integers. We consider a relative version: if the polynomials are relatively prime and no prime number divides all their values at integers, then they assume relatively prime values at at least one integer. We extend the question to all integral domains and prove it for a number of them: PIDs, UFDs containing an infinite field, polynomial rings over a UFD. Applications include a new "integral" version of the Hilbert Irreducibility Theorem, for which the irreducibility conclusion is over the ring.