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 F-1,..., F-m is an element of F-q[t][x], we show that the number off integral is an element of q[t] of degree n >= max(3, deg(t) F-1, ..., deg(t) F-m) such that all F-i(t, integral) is an element of F-q[t], 1 <= i <= m, in, are irreducible is (Pi(m)(i=1) mu(i)/N-i)q(n+1)(1 + O-m, (max deg) F-i,F- n(q(-1/2))), where N-i = n deg(x) F-i is the generic degree of F-i(t, f) for deg f = n and mu(i), is the number of factors into which F-i splits over (F) over bar (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 F-1, ... , 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 Pi(m)(i=1) F-i(t, x) = 0 (this number is always bounded above by (Sigma(m)(i=1) deg Fi)(2)/2, where deg denotes the total degree in t, x) and P = char F-q > max(1 <= i <= m) N-i, where N-i is the generic degree of F-i(t, f) for deg f = n.