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Abstrakty
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.
$q_{p,w}(u) ≡ (u^w-u^{wp})/p mod p$ 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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
121-134
Opis fizyczny
Daty
wydano
2015
Twórcy
autor
- Provincial Key Laboratory of, Applied Mathematics, Putian University, Putian, Fujian 351100, P.R. China
autor
- Johann Radon Institute for, Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-2-2