Sums of three cubes over a function field

TL;DR

This paper applies a function field circle method to show that a positive proportion of polynomials over finite fields can be expressed as sums of three cubes, under certain conjectural and characteristic assumptions.

Contribution

It introduces a function field analogue of the circle method to prove sums of three cubes representation results, assuming a form of the Ratios Conjecture.

Findings

A positive proportion of polynomials are sums of three cubes.

The method relies on a function field version of the circle method.

Results depend on a conjectural assumption related to the Ratios Conjecture.

Abstract

We use a function field version of the circle method to prove that a positive proportion of elements in $\mathbb{F}_q[t]$ are representable as a sum of three cubes of minimal degree from $\mathbb{F}_q[t]$, assuming a suitable form of the Ratios Conjecture and that the characteristic is greater than 3. The analogue of this conjecture for quadratic Dirichlet $L$-functions is known for large fixed $q$, via recent developments in homological stability.