On the GCD of shifted polynomial powers, iterations and their relatives

Let $a,b$ be multiplicatively independent positive integers and $\varepsilon>0$.  Bugeaud, Corvaja and Zannier (2003)
proved that $\gcd(a^n-1,b^n-1)\le \exp(\varepsilon n)$ for sufficiently large $n$. Ailon and Rudnick (2004) considered
the function field analogue and proved a much stronger result. In this talk we present several extensions of the result of
Ailon and Rudnick. We also look at some gcd problems for linear recurrence sequences, posing some open questions,
and if time allows on compositional iterates of univariate polynomials.