An explicit approach to Hypothesis H for polynomials over a finite field

Abstract. Schinzel’s Hypothesis H predicts that a family of irre-ducible polynomials over the integers satisfying certain necessary local conditions simultaneously assumes prime values infinitely of-ten. Here we consider an analogue of Hypothesis H for one-variable polynomials over the q-element finite field Fq and show that it holds whenever q is large compared to the degree of the product of the polynomials involved. We also show that for fixed q, the conclu-sion of our Hypothesis H holds for “almost all ” single-polynomial families. Along the way we propose a new polynomial analogue of the Hardy-Littlewood/Bateman-Horn conjectures