Algebro-Geometric Aspects of Analytic Combinatorics in Several Variables.
Stephen Melczer, SFU
Thursday May 25rd at 11:30am in K9509.
Abstract In the study of enumeration, a standard method of deriving asymptotic information about a sequence of interest is to study analytic properties of its generating function. In many interesting applications, however, one does not have access to the univariate generating function directly but knows it as a sub-series of the multivariate power series expansion of a rational function in several variables. Trying to determine asymptotic information from this less direct representation is the purpose of the new field of Analytic Combinatorics in Several Variables (ACSV), which raises many interesting questions across several domains of mathematics. This talk will discuss the algebro-geometric problems which arise, and detail the first rigorous algorithms implementing the methods of ACSV under assumptions which are often satisfied in practice. This work combines effective versions of the arithmetic Nullstellensatz with tools from polynomial system solving dating back to Kronecker to design rigorous symbolic-numeric algorithms. Genericity of the required assumptions is proven through the use of the multivariate resultant.