  Sequences that are defined by multisums of hypergeometric terms with compact
support occur frequently in enumeration problems of combinatorics, algebraic
geometry and perturbative quantum field theory. The standard recipe to study
the asymptotic expansion of such sequences is to find a recurrence satisfied by
them, convert it into a differential equation satisfied by their generating
series, and analyze the singulatiries in the complex plane. We propose a
shortcut by constructing directly from the structure of the hypergeometric term
a finite set, for which we conjecture (and in some cases prove) that it
contains all the singularities of the generating series. Our construction of
this finite set is given by the solution set of a balanced system of polynomial
equations of a rather special form, reminiscent of the Bethe ansatz. The finite
set can also be identified with the set of critical values of a potential
function, as well as with the evaluation of elements of an additive $K$-theory
group by a regulator function. We give a proof of our conjecture in some
special cases, and we illustrate our results with numerous examples.
