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$\deg f=n$.