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 $$\left(\prod_{i=1}^m\frac{μ_i}{N_i}\right) q^{n+1}\left(1+O_{m,\,\max\mathrm{deg} F_i,\,n}\left(q^{-1/2}\right)\right),$$ where $N_i=n\mathrm{deg}_xF_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg} f=n$ and $μ_i$ 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 $\mathrm{deg} f=n$.
The Multidimensional Bateman Horn Conjecture: Effective Estimates for the Tail of the Singular Series and an Algorithm for Computation with Guaranteed Accuracy
Bateman-Horn, polynomial Chowla and the Hasse principle with probability 1
IST Austria（奥地利科学技术研究所）；Dipartimento di Matematica Università di Roma Tor Vergata（罗马托维加塔大学数学系）；Department of Pure Mathematics and Mathematical Statistics University of Cambridge（剑桥大学纯数学与数理统计系）
Stellenbosch University（斯泰伦博斯大学）；University of Oxford（牛津大学）；Graz University of Technology（格拉茨工业大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件研究所）
The distribution of prime values of random polynomials
St John's College, Oxford（牛津大学圣约翰学院）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；Stanford University（斯坦福大学）；Courant Institute of Mathematical Sciences（纽约大学柯朗数学科学研究所）
