Polynomial products modulo primes and applications

Abstract

For any polynomial P (x) ∈ Z[x], we study arithmetic dynamical systems generated by F P (n) = k≤n P (n)(mod p), n ≥ 1.We apply this to improve the lower bound on the number of distinct quadratic fields of the form Q( F P (n)) in short intervals M ≤ n ≤ M +H previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of F P (n)(mod p) for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.