Greatest common divisors and Vojta’s conjecture for blowups of algebraic tori

We give results and inequalities bounding the greatest common divisor of multivariable polynomials evaluated at S -unit arguments, generalizing to an arbitrary number of variables results of Bugeaud–Corvaja–Zannier, Hernández–Luca, and Corvaja–Zannier. In closely related results, and in line with observations of Silverman, we prove special cases of Vojta’s conjecture for blowups of toric varieties. As an application, we classify when terms from simple linear recurrence sequences can have a large greatest common divisor (in an appropriate sense). The primary tool used in the proofs is Schmidt’s Subspace Theorem from Diophantine approximation.