The Incongruence of Consecutive Values of Polynomials

Suppose thatf x. PutDf(n) = min{k> 0 f(1), . . . ,f(n) are pairwise incongruent modulok}. Special cases of this function were previously considered, using methods from elementary number theory. Results from the theory of finite fields are used to prove a theorem that for allfin a large subset of x provides a characterization ofDf(n) for allnsufficiently large. This theorem partially encompasses results due to Bremser, Schumer and Washington and to Moree and Mullen, who characterizedDf(n) for cyclic, respectively, Dickson polynomials.