On some extensions of the Ailon-Rudnick Theorem
Talk by Prof. Dr. Alina Ostafe
Speaker invited by: Prof. Dr. Joachim Rosenthal
Date: 16.08.16 Time: 14.00 - 15.00 Room:
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) $$ for a sufficiently large $n$. Moreover, Ailon and Rudnick conjectured that when $\gcd(a-1,b-1)=1$, then $\gcd(a^n-1,b^n-1)=1$ infinitely often. Using finiteness of the number of torsion points on curves, Ailon and Rudnick (2004) proved the function field analogue of this conjecture, in a stronger form, that is, if $f,g\in\C[X]$ are multiplicatively independent polynomials, then there exists $h \in \C[X]$ such that for all $n\ge 1$ we have $$ \gcd(f^n-1,g^n-1) \mid h. $$In this talk we present some extensions of this result, both in the univariate and multivariate cases.