University of Sydney Algebra Seminar

Alina Ostafe (University of NSW)

Tuesday 26 November, 12-1pm, Place: Carslaw 375

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, that is, if \(f,g\in\mathbb{C}[X]\) are multiplicatively independent polynomials, then there exists \(h \in \mathbb{C}[X]\) such that for all \(n\ge 1\) we have \( \gcd(f^n-1,g^n-1) \mid h. \) 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.