Polynomial products modulo primes and applications

Abstract

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, 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.

Locations

• Monatshefte für Mathematik - View
• arXiv (Cornell University) - View - PDF