The Hilbert–Schinzel specialization property

We establish a version “over the ring” of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in variables, with coefficients in , of positive degree in the last variables, we show that if they are irreducible over and satisfy a necessary “Schinzel condition”, then the first variables can be specialized in a Zariski-dense subset of in such a way that irreducibility over is preserved for the polynomials in the remaining variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first variables in , 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 , e.g. UFDs and Dedekind domains.