Irreducible Polynomials with Varying Constraints on Coefficients

We study the number of prime polynomials of degree $n$ over ${\mathbb{F}}_q$ in which the $i^{th}$ coefficient is either preassigned to be $a_i \in $\mathbb{F}$_q$oroutsideasmallset$S_i \subset $\mathbb{F}$_q$. This serves as a function field analogue of a recent work of Maynard, which counts integer primes that do not have specific digits in their base-$q$ expansion. Our work relates to Pollack’s and Ha’s work, which count the amount of prime polynomials with $\ll \sqrt{n}$ and $\ll n$ preassigned coefficients, respectively. Our result demonstrates how one can prove asymptotics of the number of prime polynomials with different types of constraints to each coefficient.