# Exact solutions to Waring's problem for finite fields

@article{Winterhof2008ExactST, title={Exact solutions to Waring's problem for finite fields}, author={Arne Winterhof and Christiaan E. van de Woestijne}, journal={Acta Arithmetica}, year={2008}, volume={141}, pages={171-190} }

with xi ∈ Fq, i.e., as a sum of kth powers of elements of Fq. We can then define the Waring function g(k, q) as the maximal number of summands needed to express all elements of Fq as sums of kth powers. We note that, by an easy argument, we have g(k, q) = g(k, q), where k = gcd(k, q − 1). Hence, we will assume from now on that k divides q − 1. Several authors have established bounds on the value of g(k, q) for various choices of the parameters k and q – a survey is given in [8]. For the cases…

## 11 Citations

Finite Field Waring ’ s Problem

- 2010

In 1770, Waring asked the following question: given d ∈ N, can every positive integer can be written as a sum of a bounded number of dth powers of positive integers? We call a set A in N0 a basis of…

The Waring's number over finite fields through generalized Paley graphs.

- Mathematics
- 2019

We show that the Waring's number over a finite field $\mathbb{F}_q$, denoted $g(k,q)$, when exists, coincides with the diameter of the generalized Paley graph $\Gamma(k,q)=Cay(\mathbb{F}_{q},R_k)$…

Waring's Number in a Finite Field

- Mathematics
- 2009

Abstract Let p be a prime, n be an integer, k | pn –1, and γ(k, pn ) be the minimal value of s such that every number in 𝔽 pn is a sum of s k th powers. A known upper bound is improved to γ(k, pn )…

On the Odlyzko-Stanley enumeration problem and Waring's problem over finite fields

- Mathematics
- 2012

We obtain an asymptotic formula on the Odlyzko-Stanley enumeration problem. Let $N_m^*(k,b)$ be the number of $k$-subsets $S\subseteq F_p^*$ such that $\sum_{x\in S}x^m=b$. If $m 0$ such that |…

On the Waring problem with multivariate Dickson polynomials

- Engineering
- 2012

We extend recent results of Gomez and Winterhof, and Ostafe and Shparlinski on the Waring problem with univariate Dickson polynomials in a finite field to the multivariate case. We give some…

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

- Mathematics
- 2014

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,…

Polynomial values in affine subspaces of finite fields

- MathematicsJournal d'Analyse Mathématique
- 2019

In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known…

Waring’s Problem in Finite Fields with Dickson Polynomials

- 2010

We study the problem of finding or estimating the smallest number of summands needed to express each element of a fixed finite field as sum of values of a Dickson polynomial. We study the existence…

On the waring problem with Dickson polynomials in finite fields

- Mathematics
- 2011

We improve recent results of D. Gomez and A. Winterhof on the Waring problem with Dickson polynomials in finite fields. Our approach is based on recent advances in arithmetic combinatorics in…

INTRODUCTION TO SUM-PRODUCT PHENOMENON AND ARITHMETICAL APPLICATIONS

- 2015

This exploratory note is an introduction to additive combinatorics, with an emphasis on sum-product phenomenon and some arithmetical applications. Most results and proofs can be found in [3], [5],…

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Implementation of the results of the present paper in KASH 2.x, http://www.opt.math.tugraz.at/ ̃cvdwoest/maths/leenorm.kash

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