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arXiv · 1609.01213

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

Seth Dutter·Cole Love
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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 $r$.

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Seth Dutter, Cole Love. 2016-09-05. Waring's Problem for Polynomial Rings and the Digit Sum of Exponents. https://arxiv.org/abs/1609.01213

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