The virtual periods of linear recurrence sequences in cyclotomic fields

TL;DR

This paper studies the periodic behavior of linear recurrence sequences in cyclotomic fields, establishing that after initial terms, the sequence becomes periodic and providing bounds for its period, with applications to exponential sums.

Contribution

It proves the eventual periodicity of the sequence of generating fields in cyclotomic fields and derives explicit bounds for the period, advancing understanding of recurrence sequences in algebraic number theory.

Findings

The sequence of generating fields becomes periodic after finitely many terms.

Explicit bounds for the period of the sequence are provided.

Applications to exponential sums are demonstrated.

Abstract

A linear recurrence sequence in a cyclotomic field produces a sequence of the generating fields of each term. We show that the later sequence is periodic after removing the first finite terms, and give a bound of its period. This can be applied to exponential sums.