Small gaps between configurations of prime polynomials

TL;DR

This paper demonstrates the existence of large configurations of irreducible polynomials over finite fields separated by low degree polynomials, adapting number theory methods to polynomial settings.

Contribution

It introduces a novel adaptation of number theory techniques to construct large polynomial configurations with small gaps, extending previous integer-based results.

Findings

Existence of arbitrarily large configurations of irreducible polynomials

Construction of configurations separated by low degree polynomials

Extension of Green-Tao and Goldston-Pintz-Yıldırım methods to finite fields

Abstract

We find arbitrarily large configurations of irreducible polynomials over finite fields that are separated by low degree polynomials. Our proof adapts an argument of Pintz from the integers, in which he combines the methods of Goldston-Pintz-Y\i ld\i r\i m and Green-Tao to find arbitrarily long arithmetic progressions of generalized twin primes.