Tilings, groves and multiset permutations: asympotics of rational generating functions whose pole set includes a cone

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 cone 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.