Polynomials with minimal set of values and the equation \(f(x)=f(y)\) in a finite prime field
type: Item
instance of: scholarly article
title: Polynomials with minimal set of values and the equation \(f(x)=f(y)\) in a finite prime field
zbMATH Open document ID: 0591.12023
DOI: 10.1007/BF01137459
author: D. A. Mit'kin
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: Let p be a prime and let f(x) be a polynomial of degree n over GF(p), \(3\leq n<p\). In this paper the author studies upper bounds on \(N_ f\), the number of solutions of \(f(x)=f(y)\) in GF(p). If n \(| (p-1)\) and f(x) is of the form \(a(x+b)^ n+c\), then it is easy to see that \(N_ f=np-n+1\), which is its largest possible value. Excluding this case it is shown that (i) \(N_ f\leq np-2n+2\), (ii) \(N_ f\leq (n-1)p\) if either n \(| (p-1)\) or if \(n>4\) and \(p>(n-1)^ 2\), and (iii) there exist a constant c(n) such that \(N_ f\leq ([n/2]+1)p+c(n)\sqrt{p}\).
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 12E12
zbMATH DE Number: 3948429
zbMATH Keywords: polynomial
zbMATH Keywords: finite field
zbMATH Keywords: number of solutions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3948429