Polynomial products modulo primes and applications

Oleksiy Klurman, Marc Munsch

Research output: Contribution to journalArticleResearchpeer-review

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.

Original languageEnglish
Number of pages17
JournalMonatshefte fur Mathematik
DOIs
Publication statusE-pub ahead of print - 1 Jan 2020

    Fingerprint

Keywords

  • Diophantine equations
  • Distribution of sequences modulo p
  • Dynamical system modulo p
  • Perfect powers
  • Polynomials
  • Prime ideals of number fields

ASJC Scopus subject areas

  • Mathematics(all)

Cite this