Hebrew University topology and geometry seminar

Abstract: Let $f_1,\ldots,f_n$ denote a tuple of transcendental functions of interest and $P$ a polynomial. A multiplicity estimates is an estimate for the order of zero of $P(f_1,\ldots,f_n)$ at one or several points in terms of the degree of $P$. Such estimates play a major role in transcendental number theory. One is usually interested in functions satisfying a system of differential equations (e.g. exponential functions, elliptic functions, Eisenstein series). Most work on multiplicity estimates has been focused on algebraic elimination theoretic methods. I will discuss an alternative approach following an idea of Gabrielov, based on the study of certain appropriately chosen deformations and their Milnor fibers. We will see how many estimates obtained through algebraic methods can be understood (and improved) in terms of the cohomology of these fibers. The talk will include a historical review and all necessary background.