Polynomial products modulo primes and applications

For any polynomial \(P(x)\in {\mathbb {Z}}[x],\) we study arithmetic dynamical systems generated by \(\displaystyle {F_P(n)=\mathop {\prod \nolimits _{k\le n}}}P(k)(\text {mod}\ p),\)\(n\ge 1\). 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, Quiros 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.