Asymptotics of multivariate generating functions

Given a multivariate generating function $F(X) = F(x_1 , ... , x_d) = \sum_R a_R X^R$ how do we compute asymptotics for the coefficients $a_R$? I will discuss an approach based on Morse theory and multivariate complex analysis. This approach is outlined in several papers with M. Wilson, Y. Baryshnikov and others. In this talk, I will pay particular attention to the problem of effective computation, and discuss the progress we have made so far toward implementing an algorithm in the bivariate case ($d=2$).