Asymptotics of multivariate sequences IV: generating functions with poles on a hyperplane arrangement

Let F ( z 1 , . . . , z d ) be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we study how asymptotics change between different regimes, and report on an accompanying computer algebra package implementing our results. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists.

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