Asymptotics of coefficients of multivariate generating functions: improvements for multiple points

Let $F(x) = sum_{ uin mathbb{N}^d} F_ u x^ u$ be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume $F = G/H$ for some functions $G$ and $H$ holomorphic in a neighborhood of the origin. We derive asymptotics for the coefficients $F_{ralpha}$ as $r oinfty$ with $ralphain mathbb{N}^d$ for $alpha$ in a permissible subset of $d$-tuples of positive reals. More specifically, we give an algorithm for computing arbitrary terms of the asymptotic expansion for $F_{ralpha}$ when the asymptotics are controlled by a transverse multiple point of the analytic variety $H = 0$. This improves upon earlier work by R. Pemantle and M. C. Wilson. We have implemented our algorithm in Sage and apply it to obtain accurate nu- merical results for several rational combinatorial generating functions.