Waring's Problem for Polynomial Rings and the Digit Sum of Exponents

TL;DR

This paper explores representing elements in polynomial rings over fields of positive characteristic as sums of perfect powers, establishing bounds based on digit sums of exponents and providing specific growth estimates.

Contribution

It introduces methods to express polynomials as sums of perfect powers with bounds related to the digit sum of exponents in base-$p$ expansion.

Findings

Bounds on the number of perfect powers needed based on digit sum

For fixed prime $p>2$, the number of powers grows polynomially with $r$

Established techniques for representing polynomials as sums of perfect powers

Abstract

Let $F$ be an algebraically closed field of characteristic $p>0$. In this paper we develop methods to represent arbitrary elements of $F[t]$ as sums of perfect $k$-th powers for any $k\in\mathbb{N}$ relatively prime to $p$. Using these methods we establish bounds on the necessary number of $k$-th powers in terms of the sum of the digits of $k$ in its base-$p$ expansion. As one particular application we prove that for any fixed prime $p>2$ and any $\epsilon>0$ the number of $(p^r-1)$-th powers required is $\mathcal{O}\left(r^{(2+\epsilon)\ln(p)}\right)$ as a function of $r$.