The Hilbert–Schinzel specialization property

Abstract

Abstract We establish a version “over the ring” of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in k + n {k+n} variables, with coefficients in ℤ {\mathbb{Z}} , of positive degree in the last n variables, we show that if they are irreducible over ℤ {\mathbb{Z}} and satisfy a necessary “Schinzel condition”, then the first k variables can be specialized in a Zariski-dense subset of ℤ k {\mathbb{Z}^{k}} in such a way that irreducibility over ℤ {\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 ℤ k {\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 ℤ {\mathbb{Z}} , e.g. UFDs and Dedekind domains.