On the Bateman-Horn conjecture for polynomials over large finite fields

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 F₁,⋯ F_m ∈ F_q[t][x], we show that the number of ∈ F_q[t] degree n ≥ max(3; deg_t F₁, ⋯ deg_t F_m) such that all F_i(t; f) ∈ F_q[t]; 1 ≤ i ≤ m, are irreducible is {equation presented} where N_i = n deg_x F_i is the generic degree of F_i(t; f) for deg f = n and μ_i is the number of factors into which F_i splits over 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 F_q(t)) polynomials F₁, . . . , F_m not necessarily monic in x under the assumptions that nis greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve C defined by the equation {equation presented} (this number is always bounded above by (∑^m _i=1 deg F_i)²/2, where denotes the total degree in t; x) and {equation presented} where N_i is the generic degree of F_i(t; f) for deg f = n.