Bounded gaps between prime polynomials with a given primitive root

TL;DR

This paper proves unconditionally in the function field setting that there are bounded gaps between prime polynomials with a specified primitive root, extending classical prime gap results to a new algebraic context.

Contribution

It establishes the first unconditional proof of bounded gaps between primitive polynomials in function fields, analogous to Pollack's conditional result over number fields.

Findings

Bounded gaps between primitive polynomials in function fields are proven unconditionally.

The result applies to monic irreducible polynomials with a given primitive root.

Special case includes bounded gaps between primitive polynomials (e.g., g(t) = t).

Abstract

A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide an unconditional proof of the analogue of Pollack's work in the function field case; namely, that given a monic polynomial $g(t)$ which is not an $v$th power for any prime $v$ dividing $q-1$, there are bounded gaps between monic irreducible polynomials $P(t)$ in $\mathbb{F}_q[t]$ for which $g(t)$ is a primitive root (which is to say that $g(t)$ generates the group of units modulo $P(t)$). In particular, we obtain bounded gaps between primitive polynomials, corresponding to the choice $g(t) = t$.