Polynomial products modulo primes and applications

TL;DR

This paper investigates polynomial products modulo primes within arithmetic dynamical systems, providing new bounds on quadratic fields and analyzing missing values, thereby advancing understanding in number theory and polynomial dynamics.

Contribution

It introduces new bounds on quadratic fields generated by polynomial products and estimates missing values for specific polynomial families, extending prior research.

Findings

Improved lower bounds on the number of quadratic fields in short intervals.

Estimated average missing values of polynomial products modulo primes.

Generalized previous results to broader classes of polynomials.

Abstract

For any polynomial $P(x)\in\mathbb{Z}[x],$ we study arithmetic dynamical systems generated by $\displaystyle{F_P(n)=\prod_{k\le n}}P(n)(\text{mod}\ p),$ $n\ge 1.$ 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, Quir\'{o}s 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.