Equations in finite fields with restricted solution sets. II: Algebraic equations
type: Item
instance of: scholarly article
title: Equations in finite fields with restricted solution sets. II: Algebraic equations
zbMATH Open document ID: 1199.11141
published in: Acta Mathematica Hungarica
publication date: 2008-10-21T00:00:00Z
review text: Let \(\mathbb F_{q}\) be a finite field with \(q\) elements, and \(\mathcal A\), \(\mathcal B\), \(\mathcal C\), \(\mathcal D\) ``large'' subsets of \(\mathbb F_{q}\). Among several other results the authors prove that if \(| \mathcal A| |\mathcal B| |\mathcal C| |\mathcal D| > q^3\) (resp. \(| \mathcal A| |\mathcal B| |\mathcal C| |\mathcal D| > 100 q^3\)) then the equation \(a+b=cd\) (resp. the equation \(ab+1=cd\)) is solvable in elements \(a \in \mathcal A\), \(b \in\mathcal B\), \(c \in\mathcal C\) and \(d \in\mathcal D\). The proofs are based on character sum estimates obtained earlier in Part I [Acta Math. Hung. 118, No. 1-2, 129--148 (2008; Zbl 1164.11074)].
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11T30
zbMATH DE Number: 5355132
zbMATH Keywords: finite field
zbMATH Keywords: equation
zbMATH Keywords: character sum
author: Katalin Gyarmati
author: András Sárközy
MaRDI profile type: Publication
full work available at URL: s10474-007-7035-0
OpenAlex ID: W2021527726
cites work: On a problem of Diophantus
cites work: On sums and products of residues modulo p
cites work: Some estimates for character sums and applications
DOI: 10.1007/S10474-007-7035-0
description: scientific article; zbMATH DE number 5355132