Automatic asymptotics for coefficients of smooth, bivariate rational functions

TL;DR

This paper develops an effective, rigorous algorithm to compute the asymptotics of coefficients in bivariate rational generating functions without relying on the minimality assumption, extending previous theoretical results.

Contribution

It introduces a method to compute topological invariants and asymptotics of coefficients without the minimality hypothesis, enabling broader applicability.

Findings

Provides an effective algorithm for asymptotic coefficient computation

Removes the minimality assumption from previous asymptotic formulas

Implementation of the algorithm is in progress at a specified website

Abstract

We consider a bivariate rational generating function F(x,y) = P(x,y) / Q(x,y) = sum_{r, s} a_{r,s} x^r y^s under the assumption that the complex algebraic curve $\sing$ on which $Q$ vanishes is smooth. Formulae for the asymptotics of the coefficients a_{rs} were derived by Pemantle and Wilson (2002). These formulae are in terms of algebraic and topological invariants of the pole variety, but up to now these invariants could be computed only under a minimality hypothesis, namely that the dominant saddle lies on the boundary of the domain of convergence. In the present paper, we give an effective method for computing the topological invariants, and hence the asymptotics of the values a_{r,s}, without the minimality assumption. This leads to a theoretically rigorous algorithm, whose implementation is in progress at http://www.mathemagix.org .