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arXiv · 2609.11888

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

Matthew Hase-Liu
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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 $n\ge (2+\varepsilon)d$, the required lower bound on $q$ is polynomial in $d$ of degree $2+4/\varepsilon$.

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Matthew Hase-Liu. 2026-09-10. The asymptotic in Waring's problem over function fields beyond twice the degree. https://arxiv.org/abs/2609.11888

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