TY - JOUR

T1 - On the Bateman-Horn conjecture for polynomials over large finite fields

AU - Entin, Alexei

N1 - Publisher Copyright:
© The Author 2016.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable x) polynomials F1,⋯ Fm ∈ Fq[t][x], we show that the number of ∈ Fq[t] degree n ≥ max(3; degt F1, ⋯ degt Fm) such that all Fi(t; f) ∈ Fq[t]; 1 ≤ i ≤ m, are irreducible is {equation presented} where Ni = n degx Fi is the generic degree of Fi(t; f) for deg f = n and μi is the number of factors into which Fi splits over Fq. Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over Fq(t)) polynomials F1, . . . , Fm not necessarily monic in x under the assumptions that nis greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve C defined by the equation {equation presented} (this number is always bounded above by (∑m i=1 deg Fi)2/2, where denotes the total degree in t; x) and {equation presented} where Ni is the generic degree of Fi(t; f) for deg f = n.

AB - We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable x) polynomials F1,⋯ Fm ∈ Fq[t][x], we show that the number of ∈ Fq[t] degree n ≥ max(3; degt F1, ⋯ degt Fm) such that all Fi(t; f) ∈ Fq[t]; 1 ≤ i ≤ m, are irreducible is {equation presented} where Ni = n degx Fi is the generic degree of Fi(t; f) for deg f = n and μi is the number of factors into which Fi splits over Fq. Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over Fq(t)) polynomials F1, . . . , Fm not necessarily monic in x under the assumptions that nis greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve C defined by the equation {equation presented} (this number is always bounded above by (∑m i=1 deg Fi)2/2, where denotes the total degree in t; x) and {equation presented} where Ni is the generic degree of Fi(t; f) for deg f = n.

KW - Bateman-Horn

KW - function fields

KW - irreducible polynomials

UR - http://www.scopus.com/inward/record.url?scp=84988377943&partnerID=8YFLogxK

U2 - 10.1112/S0010437X16007570

DO - 10.1112/S0010437X16007570

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AN - SCOPUS:84988377943

SN - 0010-437X

VL - 152

SP - 2525

EP - 2544

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 12

ER -