Polynomial products modulo primes and applications

Abstract

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