The Hilbert-Schinzel specialization property

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 Z, of positive degree in the last n variables, we show that if they are irreducible over Z and satisfy a necessary "Schinzel condition", then the first k variables can be specialized in a Zariski-dense subset of Z k in such a way that irreducibility over 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 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 Z, e.g. UFDs and Dedekind domains for the last one.