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

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.