Polynomial products modulo primes and applications

Oleksiy Klurman; Marc Munsch

doi:10.1007/s00605-019-01359-6

Polynomial products modulo primes and applications

Oleksiy Klurman, Marc Munsch^*

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-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 language	English
Pages (from-to)	577-593
Number of pages	17
Journal	Monatshefte fur Mathematik
Volume	191
Issue number	3
Early online date	1 Jan 2020
DOIs	https://doi.org/10.1007/s00605-019-01359-6
Publication status	Published - 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

General Mathematics

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10.1007/s00605-019-01359-6

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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, Quir{\'o}s 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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AB - 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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KW - Distribution of sequences modulo p

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KW - Perfect powers

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KW - Prime ideals of number fields

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Polynomial products modulo primes and applications

Abstract

Keywords

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