  We consider a number of combinatorial problems in which rational generating
functions may be obtained, whose denominators have factors with certain
singularities. Specifically, there exist points near which one of the factors
is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the
coefficients of such a generating function. The computation requires some
topological deformations as well as Fourier-Laplace transforms of generalized
functions. We apply the results of the theory to specific combinatorial
problems, such as Aztec diamond tilings, cube groves, and multi-set
permutations.
