Tag - Bateman-Horn conjecture

The Bateman-Horn conjecture predicts the fraction of integers n such that n²+1 is prime, and makes similar predictions for polynomials of higher degree. In joint work in progress with Mark Shusterman, we prove an analogue of the n²+1 case, replacing natural numbers n with polynomials in 𝔽_q[u], which for instance counts the fraction of polynomials f such that f²+u is an irreducible polynomial. The proof combines geometric methods, unusual algebraic properties of polynomials, and some (very) classical number theory.