# Greatest common divisors with moving targets and consequences for linear recurrence sequences

## TL;DR

This paper explores bounds on the greatest common divisors of moving polynomials and algebraic linear recurrence sequences, extending Schmidt's Subspace Theorem to dynamic settings and connecting to recent research in number theory.

## Contribution

It extends Schmidt's Subspace Theorem to moving targets, providing new bounds on gcds of polynomials and recurrence sequences, advancing understanding in Diophantine approximation.

## Key findings

- Bounded the logarithmic gcd of moving polynomials evaluated at S-units.
- Extended results to algebraic linear recurrence sequences.
- Complemented recent work of Levin and others in the field.

## Abstract

We establish consequences of the moving form of Schmidt's Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving $S$-unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin and others.