Algebra Seminar: Ostafe -- On the GCD of shifted polynomial powers, iterations and their relatives

Alina Ostafe (University of New South Wales) 

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

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

Abstract: Let a,b be multiplicatively independent positive integers and epsilon>0.
Bugeaud, Corvaja and Zannier (2003) proved that 

$$ \gcd(a^n-1,b^n-1)\le \exp(\varepsilonn) $$

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 C[X] are multiplicatively
independent polynomials, then there exists h in C[X] such that for all n>0 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.