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Enumerating combinatorial classes of the complex polynomial vector fields in ℂ

Published online by Cambridge University Press:  20 February 2012

KEALEY DIAS
KEALEY DIAS*
Affiliation:
Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany (email: [email protected])
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Abstract

In order to understand the parameter space Ξd of monic and centered complex polynomial vector fields in ℂ of degree d, decomposed by the combinatorial classes of the vector fields, it is interesting to know the number of loci in parameter space consisting of vector fields with the same combinatorial data (corresponding to topological classification with fixed separatrices at infinity). This paper answers questions posed by Adam L. Epstein and Tan Lei about the total number of combinatorial classes and the number of combinatorial classes corresponding to loci of a specific (real) dimension q in parameter space, for fixed degree d; these numbers are denoted by cd and cd,q, respectively. These results are extensions of a result by Douady, Estrada, and Sentenac, which shows that the number of combinatorial classes of the structurally stable complex polynomial vector fields in ℂ of degree d is the Catalan number Cd−1. We show that enumerating the combinatorial classes is equivalent to a so-called bracketing problem. Then we analyze the generating functions and find closed-form expressions for cd and cd,q, and we furthermore make an asymptotic analysis of these sequences for d tending to ∞. These results are also applicable to special classes of quadratic and Abelian differentials and singular holomorphic foliations of the plane.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 416 - 440
Copyright
©2012 Cambridge University Press

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References

[1]Andronov, A. A., Leontovich, E. A., Gordon, I. I. and Maier, A. G.. Qualitative Theory of Second-Order Dynamic Systems. Wiley, New York, 1973, Nauka, Moscow, 1967, English translation.Google Scholar
[2]Bender, E. A.. Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory Ser. A 15 (1973), 91–111.Google Scholar
[3]Branner, B. and Dias, K.. Classification of polynomial vector fields in one complex variable. J. Difference Equ. Appl. 16(5) (2010), 463–517.Google Scholar
[4]Comtet, L.. Advanced Combinatorics. D. Reidel, Dordrecht, Holland, 1974.Google Scholar
[5]Davis, T.. Catalan numbers. http://geometer.org/mathcircles/catalan.pdf.Google Scholar
[6]Douady, A., Estrada, F. and Sentenac, P.. Champs de vecteurs polynomiaux sur ℂ, unpublished manuscript.Google Scholar
[7]Flajolet, P. and Noy, M.. Analytic combinatorics of non-crossing configurations. Discrete Math. 204 (1999), 203–229.CrossRefGoogle Scholar
[8]Flajolet, P. and Odlyzko, A.. Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2) (1990), 216–240.Google Scholar
[9]Flajolet, P. and Sedgewick, R.. Analytic Combinatorics. Cambridge University Press, 2009.Google Scholar
[10]Graham, R. L., Knuth, D. E. and Patashnik, O.. Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading, MA, 1994.Google Scholar
[11]Neumann, D.. Classification of continuous flows on 2-manifolds. Proc. Amer. Math. Soc. 48(1) (1975), 73–81.Google Scholar
[12]Salvy, B. and Zimmerman, P.. GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Software 20(2) (1994), 163–167.Google Scholar