Polynomial quotients: Interpolation, value sets and Waring's problem
Volume 170 / 2015
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.
