Green-Tao theorem in function fields

TL;DR

This paper extends the Green-Tao theorem to polynomials over finite fields, demonstrating that irreducible polynomials contain arbitrarily long arithmetic progressions with specific polynomial configurations.

Contribution

It adapts the Green-Tao proof from primes to the setting of polynomials over finite fields, establishing new polynomial progression results.

Findings

Irreducible polynomials contain polynomial configurations of the form {f + P g : deg(P) < k}

The proof adapts techniques from number theory to finite fields

Configurations exist for all degrees k in the polynomial setting

Abstract

We adapt the proof of the Green-Tao theorem on arithmetic progressions in primes to the setting of polynomials over a finite field, to show that for every $k$, the irreducible polynomials in $\mathbf{F}_q[t]$ contain configurations of the form $\{f+ Pg : \d(P)<k \}, g \neq 0$.