In 2003, Bugeaud, Corvaja, and Zannier gave an (essentially sharp) upper bound for the greatest common divisor $gcd(a^n-1,b^n-1)$, where $a$ and $b$ are fixed integers and $n$ varies over the positive integers. The proof required the deep Schmidt subspace theorem from Diophantine approximation. In subsequent work, Corvaja and Zannier generalized the result to the quantity $gcd(f(u,v),g(u,v))$, where $f$ and $g$ are polynomials satisfying appropriate natural conditions and $u$ and $v$ vary over a group of $S$-units in a number field. We will discuss a generalization of this result to polynomials in an arbitrary number of variables. Following an observation of Silverman, we will also explain how these results are closely connected to certain cases of Vojta's conjecture on blowups of projective space.