The number of directions determined by a function over a finite field.
type: Item
instance of: scholarly article
title: The number of directions determined by a function over a finite field.
zbMATH Open document ID: 1045.51004
author: Simeon Ball
published in: Journal of Combinatorial Theory. Series A
publication date: 2004-01-26T00:00:00Z
review text: Let \(GT(q)\) be the finite field with \(q= p^n\) elements, \(p\) prime. Let \(f\) be a function from \(GT(q)\) to \(GT(q)\), let \[ D= \Biggl\{{f(y)- f(x)\over y-x}\,\Biggl|\,x,y\in GF(q),\, x\neq y\Biggr\} \] be the set of directions determined by the function \(f\) and \(N=| D|\). The objective of this article is to prove that number \(N\) of directions determined by a function over \(GT(q)\) is either 1, at least \((q+3)/2\), or between \(q/s+1\) and \((q-1)/(s-1)\) for some \(s\) where \(GT(s)\) is a subfield of \(GT(q)\). The fact that if \(s> 2\), then the graph of \(f\) is \(GT(s)\)-linear, is proved too.
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 2032718
zbMATH Keywords: Lacunary polynomials
zbMATH Keywords: Permutation polynomials
zbMATH Keywords: Blocking sets
reviewed by: Pavel Burda
MaRDI profile type: Publication
full work available at URL: j.jcta.2003.09.006
OpenAlex ID: W2004755185
cites work: On the number of slopes of the graph of a function defined on a finite field
cites work: The number of directions determined by a function \(f\) on a finite field
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cites work: Q3319331
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DOI: 10.1016/J.JCTA.2003.09.006
description: scientific article; zbMATH DE number 2032718