On systems of linear and diagonal equation of degree \(p^i+1\) over finite fields of characteristic \(p\)
type: Item
instance of: scholarly article
title: On systems of linear and diagonal equation of degree \(p^i+1\) over finite fields of characteristic \(p\)
zbMATH Open document ID: 1163.11082
author: Francis N. Castro
author: Ivelisse M. Rubio
author: Puhua Guan
author: Raúl Figueroa
published in: Finite Fields and their Applications
publication date: 2008-08-27T00:00:00Z
review text: The object of the paper is to compute an analogue of the ``g'' of the waring problem over finite fields in a special case. More precisely, given positive integers \(k,d\) and a power \(q\) of a prime \(p.\) Let \(G = g(k,d,q)\) be the smallest positive integer such that given two elements \(a,b\) of \(\mathbb{F}_q\) one of them is a sum of \(G\) \(k\)-th powers while the other is a sum of \(G\) \(d\)-th powers. In the special case when \(k=1\) and \(d-1\) is a \(p\)-th power they got the value of \(G.\) Namely, \(p\) odd implies \(G=3\) iff \(q \neq (d-1)^2.\) The study of the particular polynomial \(P =(ax+b)x^q+(bx+d)\) is key. Should be interesting to settle the case \(p=2.\) Moreover, we may wonder what happens when \(k>1\).
reviewed by: Luis Gallardo
Mathematics Subject Classification ID: 11T30
Mathematics Subject Classification ID: 11D79
zbMATH DE Number: 5316919
zbMATH Keywords: system of diagonal equations
zbMATH Keywords: finite fields
zbMATH Keywords: variant of Waring's problem
MaRDI profile type: Publication
full work available at URL: j.ffa.2007.09.008
OpenAlex ID: W2005722215
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DOI: 10.1016/J.FFA.2007.09.008
description: scientific article; zbMATH DE number 5316919