Polynomials with minimal set of values and the equation \(f(x)=f(y)\) in a finite prime field (Q1074664)

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 \(N_ f\leq ([n/2]+1)p+c(n)\sqrt{p}\).