Diagonal asymptotics for symmetric rational functions via ACSV

Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle, Armin Straub

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider asymptotics of power series coefficients of rational functions of the form 1/Q where Q is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV sometimes requires considerable overhead and geometric computation, in the case of symmetric multilinear rational functions there are some reductions that streamline the analysis. Our results include diagonal asymptotics across entire classes of functions, for example the general 3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where the denominator in terms of elementary symmetric functions is 1-e1 +ced in any number d of variables. The ACSV analysis also explains a discontinuous drop in exponential growth rate for the GRZ class at the parameter value c = (d - 1)d-1, previously observed for d = 4 only by separately computing diagonal recurrences for critical and noncritical values of c.

Original languageEnglish (US)
Title of host publication29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018
EditorsMark Daniel Ward, James Allen Fill
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770781
DOIs
StatePublished - Jun 1 2018
Event29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018 - Uppsala, Sweden
Duration: Jun 25 2018Jun 29 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume110
ISSN (Print)1868-8969

Other

Other29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018
CountrySweden
CityUppsala
Period6/25/186/29/18

Keywords

  • Analytic combinatorics
  • Coefficient
  • D-finite
  • Generating function
  • Lacuna
  • Morse theory
  • Positivity
  • Smooth point

ASJC Scopus subject areas

  • Software

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  • Cite this

    Baryshnikov, Y., Melczer, S., Pemantle, R., & Straub, A. (2018). Diagonal asymptotics for symmetric rational functions via ACSV. In M. D. Ward, & J. A. Fill (Eds.), 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018 [12] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 110). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.AofA.2018.12