The aim of this project is to systematize (asymptotic) coefficient extraction from a wide class of naturally occurring multivariate generating functions. We aim to take a genuinely multivariate approach, and to improve over previous work in the areas of generality, ease of use, and suitability for effective computation. The topic is inherently interdisciplinary and uses complex analysis in one and several variables, asymptotics of integrals, topology, algebraic geometry, and symbolic computation. The application areas are many: we are particularly motivated by specific naturally occurring problems arising from areas such as multivariate recurrence relations, random tilings, queueing theory, and analysis of algorithms and data structures. If you are interested in working on these problems, please let us know.

Our methods use complex analysis and oscillatory integrals to analyse singularities of explicitly known generating functions. We deal with generating functions of the form $F(\mathbf{z}) = \sum a_{r_1}\dots a_{r_d} z_1^{r_1} \cdots z_d^{r_d}$ which can locally be expressed in the form $G/H$ with $G$ and $H$ analytic functions. The singular set of this function (zero-set of $H$) plays a large role.

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Last updated 2021-11-22.