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

• Autorzy: Zhixiong Chen , Arne Winterhof
• Języki publikacji: EN

Abstrakty

EN

For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p,w}(u)$ are defined by
$q_{p,w}(u) ≡ (u^w-u^{wp})/p mod p$ with $0 ≤ q_{p,w}(u) ≤ p-1$, u ≥ 0,
which are generalizations of Fermat quotients $q_{p,p-1}(u)$.
First, we estimate the number of elements $1 ≤ u < N ≤ p$ for which $f(u)≡ q_{p,w}(u) mod p$ for a given polynomial f(x) over the finite field $𝔽_p$. 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 $𝔽_p$ 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.

Kategorie tematyczne

• 11T24: Other character sums and Gauss sums
• 11T06: Polynomials
• 11P05: Waring's problem and variants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 170
• Numer: 2
• Strony: 121-134

Daty

wydano

2015

Twórcy

autor

Zhixiong Chen

• Provincial Key Laboratory of, Applied Mathematics, Putian University, Putian, Fujian 351100, P.R. China

autor

Arne Winterhof

• Johann Radon Institute for, Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria

Identyfikatory

DOI

10.4064/aa170-2-2