Asymptotics of Multivariate Sequences, part I. Smooth points of the singular variety

TL;DR

This paper develops a method to determine the asymptotic behavior of coefficients of multivariate generating functions by analyzing smooth singular points using Cauchy's integral formula, providing a foundation for more complex cases.

Contribution

Introduces a new approach to asymptotic analysis of multivariate generating functions at smooth singular points using oscillating integrals.

Findings

Effective asymptotic formulas for coefficients near smooth singular points

Framework applicable to multivariate generating functions with smooth poles

Lays groundwork for analyzing more complex singularities

Abstract

Given a multivariate generating function F, we determine asymptotics for the coefficients. Our approach is to use Cauchy's integral formula near singular points of F, resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers will treat singular points of F where the local geometry is more complicated, and for which other methods of analysis are not known.