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}
}
• Published 2 October 2008
• Mathematics
• Acta Arithmetica
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
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
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
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
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
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
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],

References

SHOWING 1-10 OF 14 REFERENCES
A note on Waring's problem in finite fields
(1) g(k, p) exists if and only if p − 1 pd − 1 k for all d |n, d 6= n. We shall suppose from now on that g(k, p) exists. Several bounds for g(k, p) are known. For surveys see [7] and [13]. Recent
Numbers of solutions of equations in finite fields
Such equations have an interesting history. In art. 358 of the Disquisitiones [1, a], Gauss determines the Gaussian sums (the so-called cyclotomic “periods”) of order 3, for a prime of the form p =
Diameter lower bounds for Waring graphs and multiloop networks
• Computer Science, Mathematics
Discret. Math.
• 1993
We study the diameter of Waring graphs over Zp, where p is a prime, i.e., Cayley graphs on (Zp, +) with generators of the mth powers. For fixed degree k and large p, we obtain a lower bound of order
Algebraic coding theory
• E. Berlekamp
• Computer Science
McGraw-Hill series in systems science
• 1968
This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering
On the covering radius of cyclic linear codes and arithmetic codes
• T. Helleseth
• Computer Science, Mathematics
Discret. Appl. Math.
• 1985
The problem of finding the covering radius and minimum distance of algebraic and arithmetic codes is shown to be related to Waring's problem in a finite field and to the theory of cyclotomic numbers.
Covering radius, in: Handbook of coding theory
• Covering radius, in: Handbook of coding theory
• 1998
Implementation of the results of the present paper in KASH 2.x, http://www.opt.math.tugraz.at/ ̃cvdwoest/maths/leenorm.kash
• Arne Winterhof Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstraße
• 2009
Covering Codes
• Computer Science
North-Holland mathematical library
• 2005
On Waring's problem in finite fields, Acta Arith
• On Waring's problem in finite fields, Acta Arith
• 1998