Optimal sums of three cubes in $\mathbb{F}_q[t]$

TL;DR

This paper proves that almost all elements in the polynomial ring over a finite field can be expressed as a sum of three cubes, using the circle method under certain conjectural and characteristic assumptions.

Contribution

It extends the circle method to function fields, achieving a full asymptotic formula for sums of three cubes in $\

Findings

Density 1 of elements are representable as sums of three cubes

Achieved an asymptotic formula conjectured by Hooley in number fields

Relied on the Ratios Conjecture and characteristic > 3

Abstract

We use the circle method to prove that a density 1 of elements in $\mathbb{F}_q[t]$ are representable as a sum of three cubes of essentially minimal degree from $\mathbb{F}_q[t]$, assuming the Ratios Conjecture and that the characteristic is bigger than 3. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.