Contemporary Mathematics Analytic combinatorics in d variables : An overview

Let F (Z) = P r arZ r be a rational generating function in the d variables Z1, . . . , Zd. Asymptotic formulae for the coefficients ar may be obtained via Cauchy’s integral formula in Cd. Evaluation of this integral is not as straightforward as it is in the univariate case. This paper discusses geometric techniques that are needed for evaluation of these integrals and surveys classes of functions for which these techniques lead to explicit and effectively computable asymptotic formulae.

[1]  J. Mather.  Notes on Topological Stability , 2012 .

[2]  J. M. Boardman,et al.  Singularities of differentiable maps , 2011 .

[3]  Mark C. Wilson,et al.  Asymptotic expansions of oscillatory integrals with complex phase , 2009, 0903.3585.

[4]  Timothy DeVries.  A case study in bivariate singularity analysis , 2010 .

[5]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[6]  Mark C. Wilson,et al.  A New Approach to Asymptotics of Maclaurin Coefficients of Algebraic Functions , 2008, 1202.3826.

[7]  Mark C. Wilson,et al.  Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions , 2005, SIAM Rev..

[8]  David E. Speyer,et al.  An arctic circle theorem for Groves , 2004, J. Comb. Theory, Ser. A.

[9]  Mark C. Wilson,et al.  Asymptotics of Multivariate Sequences II: Multiple Points of the Singular Variety , 2004, Combinatorics, Probability and Computing.

[10]  Yuliy Baryshnikov,et al.  Convolutions of inverse linear functions via multivariate residues , 2004 .

[11]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[12]  Thorsten Theobald,et al.  Computing Amoebas , 2002, Exp. Math..

[13]  Mark C. Wilson,et al.  Asymptotics of Multivariate Sequences: I. Smooth Points of the Singular Variety , 2002, J. Comb. Theory, Ser. A.

[14]  Andris Ambainis,et al.  One-dimensional quantum walks , 2001, STOC '01.

[15]  Robin Pemantle,et al.  Generating functions with high-order poles are nearly polynomial , 2000 .

[16]  R. Range.  Holomorphic Functions and Integral Representations in Several Complex Variables , 1998 .

[17]  J. Propp,et al.  Local statistics for random domino tilings of the Aztec diamond , 1996, math/0008243.

[18]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[19]  Aharonov,et al.  Quantum random walks. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[20]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[21]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[22]  R. Ellis,et al.  Large deviations and statistical mechanics , 1985 .

[23]  Alexander Varchenko,et al.  Newton polyhedra and estimation of oscillating integrals , 1976 .

[24]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

[25]  M. Atiyah,et al.  Lacunas for hyperbolic differential operators with constant coefficients I , 1970 .

[26]  G. Shilov,et al.  Generalized Functions, Volume 1: Properties and Operations , 1967 .

[27]  M. Riesz.  L'intégrale de Riemann-Liouville et le problème de Cauchy , 1949 .

[28]  E. Meinrenken.  Eckhard Meinrenken Clifford Algebras and Lie Theory ( Ergebnisse Der Mathematik Und Ihrer Grenzgebiete , 2022 .