Irreducible values of polynomials

Lior Bary-Soroker

doi:10.1016/j.aim.2011.10.006

Irreducible values of polynomials

Research output: Contribution to journal › Article › peer-review

Abstract

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.

Original language	English
Pages (from-to)	854-874
Number of pages	21
Journal	Advances in Mathematics
Volume	229
Issue number	2
DOIs	https://doi.org/10.1016/j.aim.2011.10.006
State	Published - 30 Jan 2012
Externally published	Yes

Keywords

Bateman-Horn conjecture
Hilbert's irreducibility theorem
Irreducible polynomials
Pseudo algebraically closed fields
Schinzel's Hypothesis H

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1016/j.aim.2011.10.006

Cite this

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title = "Irreducible values of polynomials",

abstract = "Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem {\`a} la Hilbert.",

keywords = "Bateman-Horn conjecture, Hilbert's irreducibility theorem, Irreducible polynomials, Pseudo algebraically closed fields, Schinzel's Hypothesis H",

author = "Lior Bary-Soroker",

note = "Funding Information: Part of this work was done while the author was a Lady Davis postdoc fellow in the Hebrew University of Jerusalem. While writing this work the author was an Alexander von Humboldt postdoc fellow in the Institut f{\"u}r Experimentelle Mathematik in Duisburg-Essen University. This research is partially supported by a grant from the ERC.",

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N1 - Funding Information: Part of this work was done while the author was a Lady Davis postdoc fellow in the Hebrew University of Jerusalem. While writing this work the author was an Alexander von Humboldt postdoc fellow in the Institut für Experimentelle Mathematik in Duisburg-Essen University. This research is partially supported by a grant from the ERC.

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Y1 - 2012/1/30

N2 - Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.

AB - Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.

KW - Bateman-Horn conjecture

KW - Hilbert's irreducibility theorem

KW - Irreducible polynomials

KW - Pseudo algebraically closed fields

KW - Schinzel's Hypothesis H

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DO - 10.1016/j.aim.2011.10.006

M3 - مقالة

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VL - 229

SP - 854

EP - 874

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 2

ER -

Irreducible values of polynomials

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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