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

TL;DR

This paper investigates polynomial quotients over finite fields, providing bounds on fixed points, value set sizes, and the Waring number, advancing understanding of their algebraic and additive properties.

Contribution

It introduces new bounds on value sets and fixed points of polynomial quotients and applies additive number theory to estimate the Waring number for these functions.

Findings

Bounds on the number of fixed points for polynomial quotients.

Lower bounds on the size of value sets of polynomial quotients.

Bounds on the Waring number for expressing elements as sums of polynomial quotient values.

Abstract

For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~ \mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0, $$ 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 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 from bounds on additive character sums and additive number theory.