The finite-field Bateman–Horn conjecture for one polynomial

Let $f(x)\in\mathbb{F}_q[t][x]$ be a nonconstant irreducible separable polynomial that is monic in $x$. Assume

$$\left|\\{a\in\mathbb{F}_q[t]/(P):f(a)=0\\}\right|<q^{\deg(P)}$$

for every monic irreducible $P\in\mathbb{F}_q[t]$. Define

$$C_q(f):=\prod_P\frac{1-|P|^{-1}\left|\\{\alpha\in\mathbb{F}_q[t]/(P):f(\alpha)\equiv0\pmod P\\}\right|}{1-|P|^{-1}},$$

where $|P|=q^{\deg_t P}$.

Finite-field Bateman–Horn conjecture. As $X\to\infty$ through powers of $q$,

$$\left|\\{g\in\mathbb{F}_q[t]:|g|=X,\\ g\text{ monic},\\ f(g)\in\mathcal P(\mathbb{F}_q[t])\\}\right|\approx\frac{C_q(f)}{\deg_x(f)}\frac{X}{\log_q X}.$$

This is the one-polynomial function-field analogue of Bateman–Horn, previously stated in the works cited by the source. Its general validity remains conjectural; separability is essential in the formulation.