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Polynomial quotients: Interpolation, value sets and Waring's problem

Volume 170 / 2015

Zhixiong Chen, Arne Winterhof Acta Arithmetica 170 (2015), 121-134 MSC: Primary 11P05; Secondary 11T06, 11T24. DOI: 10.4064/aa170-2-2

Abstract

For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p \ \quad \mathrm{with}\ 0 \le q_{p,w}(u) \le p-1, \,u\ge 0, $$ which are generalizations of Fermat quotients $q_{p,p-1}(u)$.

First, we estimate the number of elements $1\le u< N\le p$ for which $f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the number of fixed points of polynomial quotients.

Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of $\mathbb{F}_p$ as a sum of values of polynomial quotients, we prove some lower bounds on the size of their value sets, and then we apply these lower bounds to prove some bounds on the Waring number using results about bounds on additive character sums and from additive number theory.

Authors

  • Zhixiong ChenProvincial Key Laboratory of
    Applied Mathematics
    Putian University
    Putian, Fujian 351100, P.R. China
    e-mail
  • Arne WinterhofJohann Radon Institute for
    Computational and Applied Mathematics
    Austrian Academy of Sciences
    Altenberger Straße 69
    A-4040 Linz, Austria
    e-mail

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