A generating function is just a power series, whose
coefficients have some combinatorial interpretation.
If we are lucky enough to represent a combinatorial
problem by a nice (say rational) generating function, how can we get back asymptotic information about the coefficients?
In other words, given a rationial function F = P/Q, how can
we estimate its Taylor coefficients?
I will survey some recent work on this problem (joint with M. Wilson, Y. Baryshnikov and others). Highlights
are as follows. Complex analytic methods reduce the
problem to one of multivariate contour integration.
The leading term is determined by the variety, V, where
Q vanishes. The geometry of V determines limit behavior
of the coefficients of F. Near smooth points of V or
normal intersections, asymptotic formulae may be obtained
nearly automatically; in most cases the correct formula may
be obtained, though in general, topological computations may be required that are not known to be effective. Near other singularities of V, more difficult meromorphic integrals
are required.