Transcript seminario di fisica - Scuola Normale Superiore

SEMINARIO DI MATEMATICA
Martedì 6 dicembre 2016
ore 16:00
Scuola Normale Superiore
Pisa
Sala degli Stemmi
Alina Ostafe
(University of UNSW, Australia)
terrà un seminario dal titolo:
“On the GCD of shifted polynomial powers and
iterations”
Abstract:
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)
$$
for a sufficiently large $n$. Ailon and Rudnick (2004) considered the function field analogue and proved
a much stronger result, that is, if $f,g\in\C[X]$ are multiplicatively independent polynomials, then there
exists $h \in \C[X]$ such that for all $n\ge 1$ we have
$$
\gcd(f^n-1,g^n-1) \mid h.
$$
In this talk we present some extensions of the result of Ailon and Rudnick, both in the univariate and
multivariate cases. These extensions rely on results on the intersection of curves with algebraic subgroups
of codimension at least 2, Hilbert's irreducibility theorem and a result of Granville and Rudnick about
torsion points on hypersurfaces. We also look at some variants of the Ailon-Rudnick result for iterations
of polynomials.
Piazza dei Cavalieri, 7
56126 Pisa-Italy
tel: +39 050 509111
fax: +39 050 563513
29/11/2016