Abstract:
Let [see equation in PDF] be a multivariate power series. For example
[ibid] could
be a generating function for a combinatorial class. Assume that in a neighbourhood
of the origin this series represents a nonentire function F = G/Hp where G and H
are holomorphic and p is a positive integer. Given a direction α∈N d+ for which the
asymptotics are controlled by a smooth point of the singular variety H = 0, we compute
the asymptotics of Fnα as n→∞. We do this via multivariate singularity analysis and
give an explicit formula for the full asymptotic expansion. This improves on earlier
work of R. Pemantle and the second author and allows for more accurate numerical
approximation, as demonstrated by our examples.