A polynomial analogue of the twin prime conjecture

We consider the problem of counting the number of (not necessarily monic) `twin prime pairs' $P, P+M \in \mathbf{F}_q[T]$ of degree $n$, where $M$ is a polynomial of degree $< n$. We formulate an asymptotic prediction for the number of such pairs as $q^n\to\infty$ and then prove an explicit estimate confirming the conjecture in those cases where $q$ is large compared with $n^2$. When $M$ has degree $n-1$, our theorem implies the validity of a result conditionally proved by Hayes in 1963. When $M$ has degree zero, our theorem refines a result of Effinger, Hicks and Mullen.