The Hilbert–Schinzel specialization property - Archive ouverte HAL
Article Dans Une Revue Journal für die reine und angewandte Mathematik Année : 2022

The Hilbert–Schinzel specialization property

Résumé

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.
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Dates et versions

hal-03800676 , version 1 (08-01-2024)

Identifiants

Citer

Arnaud Bodin, Pierre Dèbes, Joachim König, Salah Najib. The Hilbert–Schinzel specialization property. Journal für die reine und angewandte Mathematik, 2022, 2022 (785), pp.55-79. ⟨10.1515/crelle-2021-0083⟩. ⟨hal-03800676⟩
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