Waring's number for large subgroups of $\mathbb Z_{p}^{*}$

Todd Cochrane ¹   ; Derrick Hart ²   ; Christopher Pinner ¹   ; Craig Spencer ¹

¹ Department of Mathematics Kansas State University Manhattan, KS 66506, U.S.A.
² Department of Mathematics Rockhurst University Kansas City, MO 64110, U.S.A.

Acta Arithmetica, Tome 163 (2014) no. 4, pp. 309-325

Résumé

Let $p$ be a prime, $\mathbb Z_p$ be the finite field in $p$ elements, $k$ be a positive integer, and $A$ be the multiplicative subgroup of nonzero $k$th powers in $\mathbb Z_p$. The goal of this paper is to determine, for a given positive integer $s$, a value $t_s$ such that if $|A|\gg t_s$ then every element of $\mathbb Z_p$ is a sum of $s$ $k$th powers. We obtain $t_4 = p^{{22/39}+\epsilon }$, $t_5= p^{{15/29}+\epsilon }$ and for $s \ge 6$, $t_s= p^{\frac {9s+45}{29s+33}+\epsilon }$. For $s \ge 24$ further improvements are made, such as $t_{32}=p^{{5/16} + \epsilon }$ and $t_{128} = p^{{1/4}}$.

Détail

DOI : 10.4064/aa163-4-2

Keywords: prime mathbb finite field elements nbsp positive integer multiplicative subgroup nonzero kth powers mathbb paper determine given positive integer value every element mathbb sum kth powers obtain epsilon epsilon frac epsilon further improvements made epsilon