We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions d at least 4. This result depends on computations in the homology of the algebraic variety where the generating function has a pole. These computations are similar to, and inspired by, a thread of research in applications of complex algebraic geometry to hyperbolic PDEs, going back to Leray, Petrowski, Atiyah, Bott and Garding. As a consequence, we give a topological explanation for certain asymptotic phenomenon appearing in the combinatorics and number theory literature. Furthermore, we show how to combine topological methods with symbolic algebraic computation to determine explicitly the dominant asymptotics for such multivariate generating functions, giving a significant new tool to attack the so-called connection problem for asymptotics of P-recursive sequences. This in turn enables the rigorous determination of integer coefficients in the Morse-Smale complex, which are difficult to determine using direct geometric methods.