For any polynomial P(x) ∈ Z[x] , we study arithmetic dynamical systems generated by FP(n)=∏k≤nP(k)(modp),n≥ 1. We apply this to improve the lower bound on the number of distinct quadratic fields of the form Q(FP(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 FP(n)(modp) for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.
Polynomial products modulo primes and applications
Munsch M.
2020-01-01
Abstract
For any polynomial P(x) ∈ Z[x] , we study arithmetic dynamical systems generated by FP(n)=∏k≤nP(k)(modp),n≥ 1. We apply this to improve the lower bound on the number of distinct quadratic fields of the form Q(FP(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 FP(n)(modp) for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.
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