{
"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"
}
}
}