Greatest common divisors for polynomials in almost units and applications to linear recurrence sequences

TL;DR

This paper establishes bounds on the greatest common divisors of polynomial evaluations at algebraic numbers, extending previous work and applying these results to linear recurrence sequences.

Contribution

It generalizes bounds for gcd of multivariable polynomials evaluated at algebraic numbers and applies these to analyze gcd of terms in linear recurrence sequences.

Findings

Bounded gcd of polynomial evaluations at algebraic numbers

Extended results to linear recurrence sequences

Improved upon previous bounds by Levin, Grieve, and Wang

Abstract

We bound the greatest common divisor of two coprime multivariable polynomials evaluated at algebraic numbers, generalizing work of Levin, and going towards conjectured inequalities of Silverman and Vojta. As an application, we prove results on greatest common divisors of terms from two linear recurrence sequences, extending the results of Levin, who considered the case where the linear recurrences are simple, and improving recent results of Grieve and Wang. The proofs rely on Schmidt's Subspace Theorem.