  The coefficient sequences of multivariate rational functions appear in many
areas of combinatorics. Their diagonal coefficient sequences enjoy nice
arithmetic and asymptotic properties, and the field of analytic combinatorics
in several variables (ACSV) makes it possible to compute asymptotic expansions.
We consider these methods from the point of view of effectivity. In particular,
given a rational function, ACSV requires one to determine a (generically)
finite collection of points that are called critical and minimal. Criticality
is an algebraic condition, meaning it is well treated by classical methods in
computer algebra, while minimality is a semi-algebraic condition describing
points on the boundary of the domain of convergence of a multivariate power
series. We show how to obtain dominant asymptotics for the diagonal coefficient
sequence of multivariate rational functions under some genericity assumptions
using symbolic-numeric techniques. To our knowledge, this is the first
completely automatic treatment and complexity analysis for the asymptotic
enumeration of rational functions in an arbitrary number of variables.
