Integrally Hilbertian rings and the polynomial Schinzel hypothesis
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911574764027904 |
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| author | Behajaina, Angelot Dèbes, Pierre König, Joachim |
| author_facet | Behajaina, Angelot Dèbes, Pierre König, Joachim |
| contents | The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the notion of {integrally Hilbertian rings}, where specialization takes place inside a ring and irreducibility is required over the ring. A core part shows how new obstacles to irreducibility such as coefficient divisors or fixed divisors can be dealt with over Krull domains, a large class of rings including UFDs, Dedekind domains, etc. As a result, we obtain a general criterion for integral hilbertianity, along with many examples, \hbox{e.g.} all rings of integers of number fields. Polynomial rings over arbitrary domains are other examples. As an application, we prove a polynomial variant of the Schinzel Hypothesis on prime values of polynomials with integer coefficients: if $\mathcal{Z}$ is an integrally Hilbertian ring, the hypothesis becomes a true statement if the ring of integers ${\mathbb Z}$ is replaced by the polynomial ring $\mathcal{Z}[U]$ and ``prime'' by ``irreducible''. This result generalizes previous works and fits in a unified framework for Schinzel-type phenomena that we introduce. We further obtain an additional conclusion that has some noteworthy consequences for the classical Schinzel Hypothesis itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09959 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrally Hilbertian rings and the polynomial Schinzel hypothesis Behajaina, Angelot Dèbes, Pierre König, Joachim Number Theory 12E05, 12E25, 12E30 The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the notion of {integrally Hilbertian rings}, where specialization takes place inside a ring and irreducibility is required over the ring. A core part shows how new obstacles to irreducibility such as coefficient divisors or fixed divisors can be dealt with over Krull domains, a large class of rings including UFDs, Dedekind domains, etc. As a result, we obtain a general criterion for integral hilbertianity, along with many examples, \hbox{e.g.} all rings of integers of number fields. Polynomial rings over arbitrary domains are other examples. As an application, we prove a polynomial variant of the Schinzel Hypothesis on prime values of polynomials with integer coefficients: if $\mathcal{Z}$ is an integrally Hilbertian ring, the hypothesis becomes a true statement if the ring of integers ${\mathbb Z}$ is replaced by the polynomial ring $\mathcal{Z}[U]$ and ``prime'' by ``irreducible''. This result generalizes previous works and fits in a unified framework for Schinzel-type phenomena that we introduce. We further obtain an additional conclusion that has some noteworthy consequences for the classical Schinzel Hypothesis itself. |
| title | Integrally Hilbertian rings and the polynomial Schinzel hypothesis |
| topic | Number Theory 12E05, 12E25, 12E30 |
| url | https://arxiv.org/abs/2502.09959 |