Automatic asymptotics for coefficients of smooth, bivariate rational functions

Abstract

We consider a bivariate rational generating function
\[
F(x, y) = \frac{P(x, y)}{Q(x, y)} = \sum_{r, s \geq 0} a_{r,s} x^r y^s
\]
under the assumption that the complex algebraic curve \( \mathcal{V} \) on which \( Q \) vanishes is smooth. Formulae for the asymptotics of the coefficients \( \{a_{r,s}\} \) are derived in [PW02]. These formulae are in terms of algebraic and topological invariants of \( \mathcal{V} \), but up to now these invariants could be computed only under a minimality hypothesis, namely that the dominant saddle must lie 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 {\(a_{rs}\)}, without the minimality assumption. This leads to a theoretically rigorous algorithm, whose implementation is in progress at http://www.mathemagix.org