Equidistribution of polynomial sequences in function fields: resolution of a conjecture

TL;DR

This paper proves a conjecture about the equidistribution of polynomial sequences over function fields, showing they are uniformly distributed under certain irrationality and divisibility conditions.

Contribution

It fully resolves a conjecture on the conditions for equidistribution of polynomial sequences in function fields, advancing understanding in this area.

Findings

Polynomial sequences are equidistributed when certain coefficients are irrational.

The result applies to polynomials with specific divisibility conditions.

The conjecture by the authors' group is confirmed in full.

Abstract

Let $\mathbb F_q$ be the finite field of $q$ elements having characteristic $p$, and denote by $\mathbb K_\infty=\mathbb F_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb T=\mathbb K_\infty/\mathbb F_q[t]$ of the values of polynomials $f(u)\in \mathbb K_\infty [u]$ as $u$ varies over $\mathbb F_q[t]$. Let $\mathcal K$ be a finite set of positive integers, and suppose that $\alpha_r\in \mathbb K_\infty$ for $r\in \mathcal K\cup \{0\}$. We show that the polynomial $\sum_{r\in \mathcal K\cup\{0\}}\alpha_ru^r$ is equidistributed in $\mathbb T$ whenever $\alpha_k$ is irrational for some $k\in \mathcal K$ satisfying $p\nmid k$, and also $p^vk\not\in \mathcal K$ for any positive integer $v$. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.