The Schinzel hypothesis for polynomials
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- by Arnaud Bodin, Pierre Dèbes and Salah Najib PDF
- Trans. Amer. Math. Soc. 373 (2020), 8339-8364 Request permission
Abstract:
The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is replaced by a polynomial ring and prove the Schinzel hypothesis for a wide class of them: polynomials in at least one variable over the integers, polynomials in several variables over an arbitrary field, etc. We achieve this goal by developing a version over rings of the Hilbert specialization property. A polynomial Goldbach conjecture is deduced, along with a result on spectra of rational functions.References
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Additional Information
- Arnaud Bodin
- Affiliation: Université de Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France
- MR Author ID: 649245
- ORCID: 0000-0001-9933-856X
- Email: arnaud.bodin@univ-lille.fr
- Pierre Dèbes
- Affiliation: Université de Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France
- ORCID: 0000-0001-9506-1380
- Email: pierre.debes@univ-lille.fr
- Salah Najib
- Affiliation: Laboratoire ATRES, Faculté Polydisciplinaire de Khouribga, Université Sultan Moulay Slimane, BP 145, Hay Ezzaytoune, 25000 Khouribga, Morocco
- MR Author ID: 738115
- Email: slhnajib@gmail.com
- Received by editor(s): March 25, 2019
- Received by editor(s) in revised form: September 15, 2019
- Published electronically: September 29, 2020
- Additional Notes: This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and by the ANR project “LISA” (ANR-17-CE40–0023-01).
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8339-8364
- MSC (2010): Primary 12E05, 12E25, 12E30; Secondary 11C08, 11N80, 13Fxx
- DOI: https://doi.org/10.1090/tran/8198
- MathSciNet review: 4177261