The asymptotic formula for Waring's problem in function fields

TL;DR

This paper establishes asymptotic formulas for Waring's problem over polynomial rings in finite fields, deriving bounds on the minimal number of k-th powers needed, with results depending on the characteristic and divisibility conditions.

Contribution

It provides new bounds on the minimal number of polynomial k-th powers needed in Waring's problem over finite fields, especially for k not divisible by the characteristic, improving previous exponential bounds.

Findings

Bounds on $ ilde{G}_q(k)$ are quadratic in k.

Derived minor arc bounds using Vinogradov-type estimates.

Established estimates for exceptional sets in the asymptotic formula.

Abstract

Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements, and let $p$ be the characteristic of $\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$ with the property that for all $s \geq t_0$, one has the expected asymptotic formula in Waring's problem over $\mathbb{F}_q[t]$ concerning sums of $s$ $k$-th powers of polynomials in $\mathbb{F}_q[t]$. For each $k$ not divisible by $p$, we derive a minor arc bound from Vinogradov-type estimates, and obtain bounds on $\widetilde{G}_q(k)$ that are quadratic in $k$, in fact linear in $k$ in some special cases, in contrast to the bounds that are exponential in $k$ available only when $k < p$. We also obtain estimates related to the slim exceptional sets associated to the asymptotic formula.