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

Zhixiong Chen¹ ; Arne Winterhof²

¹ Provincial Key Laboratory of Applied Mathematics Putian University Putian, Fujian 351100, P.R. China
² Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenberger Straße 69 A-4040 Linz, Austria

Acta Arithmetica, Tome 170 (2015) no. 2, pp. 121-134

Résumé

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.

Détail

DOI : 10.4064/aa170-2-2

Keywords: odd prime integer polynomial quotients defined equiv frac w u bmod quad mathrm p which generalizations fermat quotients p first estimate number elements which equiv bmod given polynomial finite field mathbb particular get bounds number fixed points polynomial quotients second before study problem estimating smallest number called waring number summands needed express each element mathbb sum values polynomial quotients prove lower bounds size their value sets apply these lower bounds prove bounds waring number using results about bounds additive character sums additive number theory