Greatest common divisors of values of integer polynomials and an application to maximal tori

We describe how to compute for two polynomials $f(X),g(X)\in \mathbb{Z}[X]$ with integer coefficients the greatest common divisors of $f(z)$ and $g(z)$ for all integers $z\in \mathbb{Z}$. As an application we determine the structures (elementary divisors) of all maximal tori in exceptional groups of Lie type.