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
$\deg f=n$
.