The Hilbert–Schinzel specialization property

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