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We consider a bivariate rational generating function

under the assumption that the complex algebraic curve on which
vanishes is smooth. Formulae for the asymptotics of the coefficients
are derived in [*R.
Pemantle and M.C. Wilson*. Asymptotics of multivariate sequences].
These formulae are in terms of algebraic and topological invariants of
, but up to now these
invariants could be computed onlyunder 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 , without the minimality
assumption. This leads to a theoretically rigorous algorithm, whose
implementation is in progress at `http://www.mathemagix.org`.

**Keywords: **rational function, generating function, Morse
theory, Cauchy integral, Fourier-Laplace integral

**A.M.S. subject classification: **05A15

**Authors: **