Value sets of Dickson polynomials over finite fields
type: Item
instance of: scholarly article
title: Value sets of Dickson polynomials over finite fields
zbMATH Open document ID: 0689.12012
DOI: 10.1016/0022-314X(88)90006-6
author: Wun-Seng Chou
author: Javier Gomez-Calderon
author: Gary L. Mullen
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Let f(x) be a polynomial over \({\mathbb F}_ q\), the finite field of \(q=p^ n\) elements. Let \(V_ f=\{f(x)|\) \(x\in {\mathbb F}_ q\}\) denote the value set of f(x) and \(| V_ f|\) denote the cardinality of \(V_ f\). It is easy to see that \(| V_ f| \geq [(q-1)/d]+1\), where \([\;]\) denotes the greatest integer function. If equality holds here, \(f(x)\) is called a ``minimal value set polynomial''. Such polynomials over finite fields have been previously studied by \textit{L. Carlitz}, \textit{D. J. Lewis}, \textit{W. H. Mills} and \textit{E. G. Straus} [Mathematika 8, 121--130 (1961; Zbl 0106.01105)] and \textit{W. H. Mills} [Pac. J. Math. 14, 225--241 (1964; Zbl 0126.03004)]. Recently, \textit{J. Gomez-Calderón} and \textit{D. J. Madden} [J. Number Theory 28, No. 2, 167--188 (1988; Zbl 0634.12014)] have considered polynomials with small, but not minimal, value sets; in particular, polynomials with \(| V_ f| \leq 2\{[q-1)/d]+1\}\). Also \textit{J. Gomez-Calderón} [Mathematika 35, No. 1, 144--148 (1988; Zbl 0691.12004)] has recently proved that if \(3\leq d<p\) and \(| V_ f| \leq [(q- 1)/d]+(2(q-1)/d^ 2)-1\), then \(| V_ f| =[(q-1)/d]+1\), i.e., \(f(x)\) is a minimal value set polynomial. At the other extreme, if \(f(x)\) is a permutation polynomial over \({\mathbb F}_ q\), then \(f(x)\) has a value set of maximum possible cardinality, i.e., \(| V_ f| =q\). \textit{R. Lidl} and \textit{H. Niederreiter} [Finite fields. Cambridge: Cambridge Univ. Press (1983; Zbl 0554.12010)] have given a comprehensive survey of the many papers written on permutation polynomials over finite fields. It is easy to show that for the power polynomials \(f(x)=x^ d\), \(| V_ f|\) depends only upon \((d,q-1)\), the g.c.d. of \(d\) and \(q-1\). More specifically, if \((d,q-1)=\delta\), then \(f(x)=x^ d\) has \(| V_ f| =(q-1)/\delta +1\). Thus, if \(d\) divides \(q-1\), then \(x^ d\) is a minimal value set polynomial while, if \((d,q-1)=1\), then \(x^ d\) is a permutation polynomial on \({\…
reviewed by: John H. Hodges
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 4128943
zbMATH Keywords: polynomials over finite fields
zbMATH Keywords: value set
zbMATH Keywords: permutation polynomial
zbMATH Keywords: power polynomials
zbMATH Keywords: Dickson polynomial
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4128943