Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density

Abstract

The Hilbert specialization property of a field, whereby irreducibility of polynomials is preserved under specialization of some of the variables, admits an integral counterpart where specialization takes place inside a ring and irreducibility is required over the ring. It is known that the ring 𝒪_K of integers of any global field K satisfies this integrally Hilbertian property. Using a different approach, we reprove this result and strengthen it by showing that the corresponding integral Hilbert subsets of 𝒪_K^k are not only Zariski-dense, as required by integral hilbertianity, but also of positive density. A central ingredient is a generalization to the ring 𝒪_K of the Ekedahl--Poonen formula originally proved for ℤ and 𝔽_q[t]. This formula, of independent interest, gives the density of tuples in 𝒪_K^k for which coprime polynomials in k variables and with coefficients in 𝒪_K take coprime values. Our generalization requires estimates for the number of points of bounded height in an arithmetic progression. We obtain such estimates as our third main result, thereby contributing to the theory of heights.

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BibTeX RIS

BibTeX

@misc{behajaina2026integral,
  title = {{Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density}},
  author = {Angelot Behajaina and Pierre Dèbes and Martin Widmer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20165},
  eprint = {2610.20165},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20165},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density
AU  - Angelot Behajaina
AU  - Pierre Dèbes
AU  - Martin Widmer
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20165
UR  - https://arxiv.org/abs/2610.20165
AB  - The Hilbert specialization property of a field, whereby irreducibility of polynomials is preserved under specialization of some of the variables, admits an integral counterpart where specialization takes place inside a ring and irreducibility is required over the ring. It is known that the ring 𝒪_K of integers of any global field K satisfies this integrally Hilbertian property. Using a different approach, we reprove this result and strengthen it by showing that the corresponding integral Hilbert subsets of 𝒪_K^k are not only Zariski-dense, as required by integral hilbertianity, but also of positive density. A central ingredient is a generalization to the ring 𝒪_K of the Ekedahl--Poonen formula originally proved for ℤ and 𝔽_q[t]. This formula, of independent interest, gives the density of tuples in 𝒪_K^k for which coprime polynomials in k variables and with coefficients in 𝒪_K take coprime values. Our generalization requires estimates for the number of points of bounded height in an arithmetic progression. We obtain such estimates as our third main result, thereby contributing to the theory of heights.
ER  -

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