Irreducible values of polynomials

Lior Bary-Soroker

doi:https://doi.org/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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https://doi.org/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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AU - Bary-Soroker, Lior

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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 - https://doi.org/10.1016/j.aim.2011.10.006

M3 - مقالة

SN - 0001-8708

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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