Polynomial products modulo primes and applications

Oleksiy Klurman, Marc Munsch*

*Corresponding author for this work

Research output: Contribution to journalArticle

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
Pages (from-to)577-593
Number of pages17
JournalMonatshefte fur Mathematik
Volume191
Issue number3
Early online date1 Jan 2020
DOIs
Publication statusPublished - 1 Mar 2020

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)

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