13/05/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Konstantin Dyakonov, ICREA e Universitat de Barcelona
Zeros of analytic functions, with or without multiplicities
The so-called abc theorem for polynomials, also known as Mason's
or Mason-Stothers' theorem, deals with nontrivial polynomial
solutions to the Diophantine equation \(a+b=c\). It provides a
lower bound on the number of distinct zeros of the polynomial abc
in terms of the degrees of \(a\), \(b\) and \(c\). We prove some
"local" \(abc\) type theorems for general analytic functions living
on a (reasonably nice) bounded domain rather than on the whole
plane. The estimates obtained are sharp, for any domain, and they
imply a generalization of the original "global" \(abc\) theorem by
a limiting argument.