On the Bateman-Horm Conjecture about Polynomial Rings

Lior Bary-Soroker; Moshe Jarden

paper

On the Bateman-Horm Conjecture about Polynomial Rings

arXiv:1203.0726

Abstract

Given a power $q$ of a prime number $p$ and "nice" polynomials $f_1,...,f_r\in\bbF_q[T,X]$ with $r = 1$ if $p = 2$ , we establish an asymptotic formula for the number of pairs $(a_1,a_2)\in\bbF_q^2$ such that $f_{1} (T, a_{1} T + a_{2}), ..., f_{r} (T, a_{1} T + a_{2})$ are irreducible in $\bbF_q[T]$. In particular that number tends to infinity with $q$ .

To be published in Muenster Journal of Mathematics