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

Zhixiong Chen, and Arne Winterhof. "Polynomial quotients: Interpolation, value sets and Waring's problem." Acta Arithmetica 170.2 (2015): 121-134. <http://eudml.org/doc/279793>.

@article{ZhixiongChen2015,
abstract = {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 $_p$. 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 $_p$ 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.},
author = {Zhixiong Chen, Arne Winterhof},
journal = {Acta Arithmetica},
keywords = {polynomial quotients; Fermat quotients; Waring problem; value set; fixed points; character sums},
language = {eng},
number = {2},
pages = {121-134},
title = {Polynomial quotients: Interpolation, value sets and Waring's problem},
url = {http://eudml.org/doc/279793},
volume = {170},
year = {2015},
}

TY - JOUR
AU - Zhixiong Chen
AU - Arne Winterhof
TI - Polynomial quotients: Interpolation, value sets and Waring's problem
JO - Acta Arithmetica
PY - 2015
VL - 170
IS - 2
SP - 121
EP - 134
AB - 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 $_p$. 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 $_p$ 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.
LA - eng
KW - polynomial quotients; Fermat quotients; Waring problem; value set; fixed points; character sums
UR - http://eudml.org/doc/279793
ER -