Skip to main content
SpringerLink
Log in

Asymptotics of Multivariate Sequences IV: Generating Functions with Poles on a Hyperplane Arrangement

  • Original Paper
  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

Let \(F(z_1,\dots ,z_d)\) be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and provide the first software package to verify and compute asymptotics in non-smooth cases of analytic combinatorics in several variables. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Algorithm 1
Algorithm 2
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. See acsvproject.com for a listing of papers in this project.

  2. The fact that the original cycle is in \({\mathcal {M}}\subseteq {\mathcal {V}}^c\) while the Morse theory is done on \({\mathcal {V}}\) will be reconciled in Sect. 4.

  3. Topological cancellation, which can also cause a drop in the limsup neighbourhood exponential rate, is harder to study. See [2] for one example of this.

  4. If \({\hat{{\textbf{r}}}}\) were a generic direction, we would be able to add all \(2^t\) fibers and use univariate residues to get rid of all \(w_j\) in the integrand denominator.

References

  1. M. F. Atiyah, R. Bott, and L. Gȧrding. Lacunas for hyperbolic differential operators with constant coefficients. I. Acta Math., 124:109–189, 1970.

    Article  MathSciNet  Google Scholar 

  2. Y. Baryshnikov, S. Melczer, and R. Pemantle. Asymptotics of multivariate sequences in the presence of a lacuna. Submitted,arxiv.org/abs/1905.04187, 2021.

  3. Y. Baryshnikov and R. Pemantle. Asymptotics of multivariate sequences, part iii: quadratic points. Adv. Math., 228:3127–3206, 2011.

    Article  MathSciNet  Google Scholar 

  4. Yuliy Baryshnikov, Stephen Melczer, and Robin Pemantle. Stationary points at infinity for analytic combinatorics. Found. Comput. Math., 22(5):1631–1664, 2022.

    Article  MathSciNet  Google Scholar 

  5. E. A. Bender and L. B. Richmond. Central and local limit theorems applied to asymptotic enumeration. ii. multivariate generating functions. J. Combin. Theory Ser. A, 34:255–265, 1983.

  6. Edward A. Bender and L. Bruce Richmond. Multivariate asymptotics for products of large powers with applications to Lagrange inversion. Electron. J. Combin., 6:Research Paper 8, 21 pp. (electronic), 1999.

  7. Andrea Bertozzi and James McKenna. Multidimensional residues, generating functions, and their application to queueing networks. SIAM Rev., 35(2):239–268, 1993.

    Article  MathSciNet  Google Scholar 

  8. T. Brylawski. The broken-circuit complex. Trans. AMS, 234(2):417–433, 1977.

    Article  MathSciNet  Google Scholar 

  9. Gilles Christol. Diagonales de fractions rationnelles et equations différentielles. In Study group on ultrametric analysis, 10th year: 1982/83, No. 2, pages Exp. No. 18, 10. Inst. Henri Poincaré, Paris, 1984.

  10. J. de Loera and B. Sturmfels. Algebraic unimodular counting. Math. Program., 96:183–203, 2003.

    Article  MathSciNet  Google Scholar 

  11. T. DeVries, J. van der Hoeven, and R. Pemantle. Automatic asymptotics for coefficients of smooth, bivariate rational functions. Online J. Anal. Comb., 6:24 pages, 2011.

  12. P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009.

  13. Zhicheng Gao and L. Bruce Richmond. Central and local limit theorems applied to asymptotic enumeration. IV. Multivariate generating functions. J. Comput. Appl. Math., 41(1-2):177–186, 1992. Asymptotic methods in analysis and combinatorics.

  14. A. Gilbert and H. Karloff. On the fractal behavior of tcp. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pages 297–306, New York, 2003. ACM Press.

  15. Mark Goresky and Robert MacPherson. Stratified Morse theory, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988.

  16. Peter Henrici. Applied and computational complex analysis. Vol. 2. Wiley Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions.

  17. Lars Hörmander. The analysis of linear partial differential operators. I, volume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1990. Distribution theory and Fourier analysis.

  18. Hsien-Kuei Hwang. Large deviations of combinatorial distributions. II. Local limit theorems. Ann. Appl. Probab., 8(1):163–181, 1998.

  19. Pierre Lairez. Computing periods of rational integrals. Math. Comp., 85(300):1719–1752, 2016.

    Article  MathSciNet  Google Scholar 

  20. Sergei K. Lando and Alexander K. Zvonkin. Graphs on surfaces and their applications, volume 141 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004. With an appendix by Don B. Zagier, Low-Dimensional Topology, II.

  21. L. Lipshitz. The diagonal of a \(D\)-finite power series is \(D\)-finite. J. Algebra, 113(2):373–378, 1988.

    Article  MathSciNet  Google Scholar 

  22. M. Lladser. Uniform formulae for coefficients of meromorphic functions in two variables. SIAM J. Dis. Math., 20(4):811–828, 2006.

    Article  MathSciNet  Google Scholar 

  23. Stephen Melczer. An Invitation to Analytic Combinatorics in One and Several Variables. Springer, New York, 2021.

    Google Scholar 

  24. Stephen Melczer and Bruno Salvy. Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems. J. Symbolic Comput., 103:234–279, 2021.

    Article  MathSciNet  Google Scholar 

  25. Marc Mezzarobba. NumGfun: a package for numerical and analytic computation with D-finite functions. pages 139–146, 2010.

  26. Marc Mezzarobba. Rigorous multiple-precision evaluation of D-finite functions in SageMath. Technical Report 1607.01967, arXiv, 2016. Extended abstract of a talk at the 5th International Congress on Mathematical Software.

  27. A. M. Odlyzko. Asymptotic enumeration methods. In Handbook of combinatorics, Vol. 1, 2, pages 1063–1229. Elsevier, Amsterdam, 1995.

  28. P. Orlik and L. Solomon. Combinatorics and topology of complements of hyperplanes. Invent. Math., 56:167–189, 1980.

    Article  ADS  MathSciNet  Google Scholar 

  29. P. Orlik and H. Terao. Commutative algebras for arrangements. Nagoya Math J., 134:65–73, 1994.

    Article  MathSciNet  Google Scholar 

  30. Peter Orlik and Hiroaki Terao. Arrangements of hyperplanes, volume 300 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992.

  31. R. Pemantle, M. C. Wilson, and S. Melczer. Analytic Combinatorics in Several Variables, Second Edition. Cambridge University Press, 2023.

  32. R. Pemantle and M. C. Wilson. Asymptotics of multivariate sequences. II. Multiple points of the singular variety. Combin. Probab. Comput., 13(4-5):735–761, 2004.

  33. R. Pemantle and M. C. Wilson. Analytic Combinatorics in Several Variables. Cambridge University Press, 2013.

  34. Robin Pemantle. Generating functions with high-order poles are nearly polynomial. In Mathematics and computer science (Versailles, 2000), Trends Math., pages 305–321. Birkhäuser, Basel, 2000.

  35. A. Raichev and M.C. Wilson. Asymptotics of coefficients of multivariate generating functions: improvements for multiple points. Online Journal of Analytic Combinatorics, 6(0), 2011.

  36. Elias M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993.

  37. Joris van der Hoeven. Fast evaluation of holonomic functions near and in regular singularities. J. Symbolic Comput., 31(6):717–743, 2001.

    Article  MathSciNet  Google Scholar 

  38. A. Varchenko and I.M. Gelfand. Combinatorics and topology of configuration of affine hyperplanes in real space. Funk. Analiz i ego Prilozh., 21:11–22, 1987.

    Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referee for their careful reading and catching of typos.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Yuliy Baryshnikov

  2. Department of Combinatorics and Optimization, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

    Stephen Melczer

  3. Department of Mathematics, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA, 19104, USA

    Robin Pemantle

Authors
  1. Yuliy Baryshnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen Melczer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robin Pemantle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen Melczer.

Additional information

Communicated by Alin Bostan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baryshnikov, Y., Melczer, S. & Pemantle, R. Asymptotics of Multivariate Sequences IV: Generating Functions with Poles on a Hyperplane Arrangement. Ann. Comb. 28, 169–221 (2024). https://doi.org/10.1007/s00026-023-00654-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-023-00654-2

Search

Navigation