Effective results on the Skolem Problem for linear recurrence sequences

TL;DR

This paper provides explicit lower bounds for the index after which all terms of certain linear recurrence sequences with algebraic numbers are non-zero, covering most sequences with rational coefficients.

Contribution

It introduces new explicit bounds for the Skolem problem in cases with dominant or two maximal modulus roots, applicable to nearly all rational coefficient sequences.

Findings

Established explicit lower bounds for non-zero terms

Covered sequences with dominant or two maximal modulus roots

Applicable to almost all sequences with rational coefficients

Abstract

In this paper, given a simple linear recurrence sequence of algebraic numbers, which has either a dominant characteristic root or exactly two characteristic roots of maximal modulus, we give some explicit lower bounds for the index beyond which every term of the sequence is non-zero. It turns out that this case covers almost all such sequences whose coefficients are rational numbers.