The asymptotic in Waring's problem over function fields beyond twice the degree

Abstract

We prove the expected asymptotic in Waring's problem over ${\mathbb{F}}_q[T]$, with a power-saving error, whenever $n>2d$, the characteristic is greater than $(d-1{)}^2$, and $q$ satisfies an explicit lower bound. This range is sharp in general for the expected asymptotic uniformly in the target polynomial. Our main new input is an aggregate minor arc estimate: we count functionals according to the codimension of their associated singular loci and use intersection theory to bound the degrees of the resulting parameter spaces. In particular, if $n\ge (2+\epsilon )d$, the required lower bound on $q$ is polynomial in $d$ of degree $2+4/\epsilon$.

29 pages