On the Bateman-Horm Conjecture about Polynomial Rings

TL;DR

This paper proves an asymptotic formula for counting pairs in finite fields that make certain polynomial evaluations irreducible, extending the Bateman-Horm conjecture to polynomial rings.

Contribution

It establishes a new asymptotic formula for the number of pairs leading to irreducible polynomial evaluations, generalizing previous conjectures.

Findings

Number of such pairs tends to infinity as q increases

Asymptotic formula derived for the count of pairs

Results hold for 'nice' polynomials over finite fields

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 $f_1(T,a_1T+a_2),...,f_r(T,a_1T+a_2)$ are irreducible in $\bbF_q[T]$. In particular that number tends to infinity with $q$.