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Waring's Problem for Polynomial Rings and the Digit Sum of Exponents | | Seth Dutter
; Cole Love
; | | Date: |
5 Sep 2016 | | 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 $kinmathbb{N}$ 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
$mathcal{O}left(r^{(2+epsilon)ln(p)}
ight)$ as a function of $r$. | | Source: | arXiv, 1609.1213 | | Services: | Forum | Review | PDF | Favorites | |
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