M\"{o}bius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields

TL;DR

This paper proves new cancellation results for trace functions over polynomial rings over finite fields, leading to progress on analogs of Chowla's and Bateman-Horn conjectures in this setting, with applications to sums over primes.

Contribution

It establishes the first nontrivial bounds for trace function sums over polynomial sequences and primes in finite fields, advancing the understanding of these conjectures in function field contexts.

Findings

Achieves near square-root cancellation in trace function sums

Proves the function field analog of Chowla's conjecture for certain cases

Provides bounds for sums over primes in $\

Abstract

We establish cancellation in short sums of certain special trace functions over $\mathbb{F}_q[u]$ below the P\'{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $\mathbb{F}_q[u]$-analog of Chowla's conjecture on cancellation in M\"{o}bius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $\mathbb{F}_q[u]$.