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7 - Overview of analytic methods for multivariate GFs

from Part III - Multivariate Enumeration

Published online by Cambridge University Press:  08 February 2024

Robin Pemantle
Affiliation:
University of Pennsylvania
Mark C. Wilson
Affiliation:
University of Massachusetts, Amherst
Stephen Melczer
Affiliation:
University of Waterloo, Ontario
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Summary

This chapter gives a high-level overview of analytic combinatorics in several variables. Stratified Morse theory reduces the derivation of coefficient asymptotics for a multivariate generating function to the study of asymptotic expansions of local integrals near certain critical points on the generating function’s singular set. Determining exactly which critical points contribute to asymptotic behavior is a key step in the analysis . The asymptotic behavior of each local integral depends on the local geometry of the singular variety, with three special cases treated in later chapters.

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Publisher: Cambridge University Press
Print publication year: 2024

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