%0 Journal Article
%T Automatic asymptotics for coefficients of smooth, bivariate rational functions
%A DeVries Timothy
%A Joris van der Hoeven
%A Robin Pemantle
%J Online Journal of Analytic Combinatorics
%D 2011
%I University of Auckland
%X We consider a bivariate rational generating function $$F(x,y) = frac{P(x,y)}{Q(x,y)} = sum_{r,sgeq 0} a_{rs}x^ry^s$$ under the assumption that the complex algebraic curve $mathcal{V}$ on which $Q$ vanishes is smooth. Formulae for the asymptotics of the coefficients $a_{rs}$ are derived in [Pemantle and Wilson, 2002]. These formulae are in terms of algebraic and topological invariants of $sing$, 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 exttt{http://www.mathemagix.org}.
%U http://analytic-combinatorics.org/index.php/ojac/article/view/38