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

Abstract

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 $F_1,\ldots,F_m\in\mathbf{F}_q[t][x]$, with $q$ odd, we show that the number of $f\in\mathbf{F}_q[t]$ of degree $n\ge\max(3,\mathrm{deg}_t F_1,\ldots,\mathrm{deg}_t F_m)$ such that all $F_i(t,f)\in\mathbf{F}_q[t],1\le i\le m$ are irreducible is $$\left(\prod_{i=1}^m\frac{\mu_i}{N_i}\right) q^{n+1}\left(1+O_{m,\,\max\mathrm{deg} F_i,\,n}\left(q^{-1/2}\right)\right),$$ where $N_i=n\mathrm{deg}_xF_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg} f=n$ and $\mu_i$ is the number of factors into which $F_i$ splits over $\bar{\mathbf{F}_q}$. Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over $\mathbf{F}_q(t)$) polynomials $F_1,\ldots,F_m$ not necessarily monic in $x$ under the assumptions that $n$ is greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve $C$ defined by the equation $$\prod_{i=1}^mF_i(t,x)=0$$ (this number is always bounded above by $\left(\textstyle\sum_{i=1}^m\mathrm{deg} F_i\right)^2/2$, where $\mathrm{deg}$ denotes the total degree in $t,x$) and $$p=\mathrm{char}\,\mathbf{F}_q>\max_{1\le i\le m} N_i,$$ where $N_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg} f=n$.