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Analytic Combinatorics in Several Variables : Effective Asymptotics and Lattice Path Enumeration

Stephen Melczer 1, 2
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : The field of analytic combinatorics, which studies the asymptotic behaviour ofsequences through analytic properties of their generating functions, has led to thedevelopment of deep and powerful tools with applications across mathematics and thenatural sciences. In addition to the now classical univariate theory, recent work in thestudy of analytic combinatorics in several variables (ACSV) has shown how to deriveasymptotics for the coefficients of certain D-finite functions represented by diagonals ofmultivariate rational functions. This thesis examines the methods of ACSV from acomputer algebra viewpoint, developing rigorous algorithms and giving the firstcomplexity results in this area under conditions which are broadly satisfied.Furthermore, this thesis gives several new applications of ACSV to the enumeration oflattice walks restricted to certain regions. In addition to proving several openconjectures on the asymptotics of such walks, a detailed study of lattice walk modelswith weighted steps is undertaken.
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Stephen Melczer. Analytic Combinatorics in Several Variables : Effective Asymptotics and Lattice Path Enumeration. Symbolic Computation [cs.SC]. Université de Lyon, 2017. English. ⟨NNT : 2017LYSEN013⟩. ⟨tel-01587716⟩

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