The Hilbert-Schinzel specialization property

TL;DR

This paper proves a version of the Hilbert Irreducibility Theorem over rings like , showing that under certain conditions, one can specialize variables to preserve irreducibility and coprimality of polynomials.

Contribution

It establishes a ring-based Hilbert Irreducibility Theorem and improves the Schinzel Hypothesis, allowing specialization that preserves irreducibility and coprimality over various rings.

Findings

Specialization preserves irreducibility under Schinzel conditions.

Coprime polynomials assume coprime values under certain conditions.

Results extend to rings like UFDs and Dedekind domains.

Abstract

We establish a version "over the ring" of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in $k+n$ variables, with coefficients in $\mathbb Z$, of positive degree in the last $n$ variables, we show that if they are irreducible over $\mathbb Z$ and satisfy a necessary "Schinzel condition", then the first $k$ variables can be specialized in a Zariski-dense subset of ${\mathbb Z}^k$ in such a way that irreducibility over ${\mathbb Z}$ is preserved for the polynomials in the remaining $n$ variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first $k$ variables in ${\mathbb Z}^k$, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a "coprime" version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials assume coprime values. We prove our results over many other rings than $\mathbb Z$, e.g. UFDs and Dedekind domains for the last one.