{ "id": "2009.07254", "version": "v1", "published": "2020-09-15T17:44:25.000Z", "updated": "2020-09-15T17:44:25.000Z", "title": "The Hilbert-Schinzel specialization property", "authors": [ "Arnaud Bodin", "Pierre Dèbes", "Joachim König", "Salah Najib" ], "comment": "21 pages", "categories": [ "math.NT" ], "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 \$\\mathbb Z\$, of positive degree in the last \$n\$ variables, we show that if they are irreducible over \$\\mathbb Z\$ and satisfy a necessary \"Schinzel condition\", then the first \$k\$ variables can be specialized in a Zariski-dense subset of \${\\mathbb Z}^k\$ in such a way that irreducibility over \${\\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 \${\\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 \$\\mathbb Z\$, e.g. UFDs and Dedekind domains for the last one.", "revisions": [ { "version": "v1", "updated": "2020-09-15T17:44:25.000Z" } ], "analyses": { "subjects": [ "12E05", "12E30", "13Fxx", "11A05", "11A41" ], "keywords": [ "hilbert-schinzel specialization property", "schinzel condition", "coprime polynomials assume coprime values", "schinzel hypothesis", "common prime number" ], "note": { "typesetting": "TeX", "pages": 21, "language": "en", "license": "arXiv", "status": "editable" } } }