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 $F_1,\dots ,F_m\in {\mathbf{F}}_q[t][x]$, with $q$ odd, we show that the number of $f\in {\mathbf{F}}_q[t]$ of degree $n\ge \max (3,{\mathrm{deg}}_t{F}_1,\dots ,{\mathrm{deg}}_t{F}_m)$ 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 $N_i=n{\mathrm{deg}}_x{F}_i$ 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 $\overline{{\mathbf{F}}_q}$. 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 $F_1,\dots ,F_m$ 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 $$\prod _{i=1}^m{F}_i(t,x)=0$$ (this number is always bounded above by ${\left({\sum }_{i=1}^m\mathrm{deg}{F}_i\right)}^2/2$, where $\mathrm{deg}$ denotes the total degree in $t,x$) and $$p=\mathrm{char}{\mathbf{F}}_q>\underset{1\le i\le m}{\max }{N}_i,$$ where $N_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg}f=n$.

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• Shifted convolution and the Titchmarsh divisor problem over F_q[t]
• Prime polynomials in short intervals and in arithmetic progressions
• The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2
• On quantitative analogues of the Goldbach and twin prime conjectures over F_q[t]

Cited by in corpus (6)

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• Prime polynomial values of quadratic functions in short intervals
• Twin prime polynomials in function field