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 $F_{1},\ldots ,F_{m}\in \mathbf{F}_{q}[t][x]$ , we show that the number of $f\in \mathbf{F}_{q}[t]$ of degree $n\geqslant \max (3,\deg _{t}F_{1},\ldots ,\deg _{t}F_{m})$ such that all $F_{i}(t,f)\in \mathbf{F}_{q}[t],1\leqslant i\leqslant m$ , are irreducible is

$$\begin{eqnarray}\displaystyle \biggl(\mathop{\prod }_{i=1}^{m}\frac{\unicode[STIX]{x1D707}_{i}}{N_{i}}\biggr)q^{n+1}(1+O_{m,\,\max \deg F_{i},\,n}(q^{-1/2})), & & \displaystyle \nonumber\end{eqnarray}$$

where $N_{i}=n\deg _{x}F_{i}$ is the generic degree of $F_{i}(t,f)$ for $\deg f=n$ and $\unicode[STIX]{x1D707}_{i}$ is the number of factors into which $F_{i}$ splits over $\overline{\mathbf{F}}_{q}$ . 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

$$\begin{eqnarray}\displaystyle \mathop{\prod }_{i=1}^{m}F_{i}(t,x)=0 & & \displaystyle \nonumber\end{eqnarray}$$

(this number is always bounded above by $(\sum _{i=1}^{m}\deg F_{i})^{2}/2$ , where $\deg$ denotes the total degree in $t,x$ ) and

$$\begin{eqnarray}\displaystyle p=\text{char}\,\mathbf{F}_{q}>\max _{1\leqslant i\leqslant m}N_{i}, & & \displaystyle \nonumber\end{eqnarray}$$

where $N_{i}$ is the generic degree of $F_{i}(t,f)$ for $\deg f=n$ .