Asymptotics of Bivariate Analytic Functions with Algebraic Singularities

TL;DR

This paper applies advanced multivariate analytic techniques to derive asymptotic formulas for coefficients of multivariate generating functions with algebraic singularities, extending prior univariate and rational function analyses.

Contribution

It adapts Pemantle and Wilson's multivariate methods to analyze algebraic singularities in multivariate generating functions, providing new asymptotic formulas.

Findings

Derived asymptotic formulas for multivariate algebraic generating functions.

Extended multivariate analysis techniques to algebraic singularities.

Bridged gap between univariate and multivariate algebraic singularity analysis.

Abstract

In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions. These same multivariate techniques can be used to analyze functions with algebraic singularities.