The lemniscate tree of a random polynomial
Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1663-1687.

To each generic complex polynomial p(z) is associated a labeled binary tree (here referred to as a “lemniscate tree”) that encodes the topological type of the graph of |p(z)|. The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the singular components of those level sets |p(z)|=t passing through a critical point.

In this paper, we ask: how many branches appear in a typical lemniscate tree? We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class of all such trees associated to a generic polynomial of fixed degree and second for the lemniscate tree arising from a random polynomial with i.i.d. zeros. From a more general perspective, these results take a first step toward a probabilistic treatment (within a specialized setting) of Arnold’s program of enumerating algebraic Morse functions.

À chaque polynôme complexe générique p(z) est associé un arbre binaire étiqueté (appelé dans cet aticle, “arbre de lemniscate”) qui code le type topologique du graphe de |p(z)|. La structure de ramification de l’arbre de lemniscate est déterminée par la configuration (c’est-à-dire la disposition dans le plan) des composants singuliers de ces ensembles de niveaux |p(z)|=t passant par un point critique.

Dans cet article, nous nous intéressons à la question suivante : combien de branches apparaissent typiquement dans un arbre de lemniscate  ? Nous répondons d’abord à cette question pour un arbre de lemniscate échantillonné uniformément dans la classe combinatoire de tous ces arbres associés à un polynôme générique de degré fixé, et ensuite pour un arbre de lemniscate résultant d’un polynôme aléatoire avec des zéros indépendants et identiquement distribués. D’un point de vue plus général, ces résultats constituent un premier pas vers un traitement probabiliste (dans un cadre spécialisé) du programme d’Arnold consistant à énumérer les fonctions algébriques de Morse.

Received:
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Accepted:
Published online:
DOI: 10.5802/aif.3377
Classification: 30C15, 60G60, 31A15, 14P25, 05A15, 60C05, 60F05
Keywords: random polynomial, binary tree, lemniscate, analytic combinatorics
Mot clés : polynôme aléatoire, arbre binaire, lemniscate, combinatoire analytique

Epstein, Michael 1; Hanin, Boris 2; Lundberg, Erik 3

1 Colorado State University Dept. of Mathematics Fort Collins, CO 80523 (USA)
2 Texas A&M University Dept. of Mathematics College Station, TX 77843 (USA)
3 Florida Atlantic University Dept. of Mathematical Sciences Boca Raton, FL 33431 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Epstein, Michael; Hanin, Boris; Lundberg, Erik. The lemniscate tree of a random polynomial. Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1663-1687. doi : 10.5802/aif.3377. https://aif.centre-mersenne.org/articles/10.5802/aif.3377/

[1] Arnold, Vladimir I Topological classification of Morse functions and generalisations of Hilbert’s 16-th problem, Math. Phys. Anal. Geom., Volume 10 (2007) no. 3, pp. 227-236 | DOI | MR | Zbl

[2] Bauer, Ingrid; Catanese, Fabrizio Generic lemniscates of algebraic functions, Math. Ann., Volume 307 (1997) no. 3, pp. 417-444 | DOI | MR | Zbl

[3] Bergeron, François; Flajolet, Philippe; Salvy, Bruno Varieties of increasing trees, Colloquium on trees in algebra and programming (1992), pp. 24-48

[4] Bóna, Miklós k-Protected vertices in binary search trees, Adv. Appl. Math., Volume 53 (2014), pp. 1-11 | DOI | MR | Zbl

[5] Bóna, Miklós Balanced vertices in labeled rooted trees (2017) (https://arxiv.org/abs/1705.09688)

[6] Bóna, Miklós; Mező, István Limiting probabilities for vertices of a given rank in rooted trees (2018) (https://arxiv.org/abs/1803.05033) | Zbl

[7] Bóna, Miklós; Pittel, Boris On a random search tree: asymptotic enumeration of vertices by distance from leaves, Adv. Appl. Probab., Volume 49 (2017) no. 3, pp. 850-876 | DOI | MR | Zbl

[8] Catanese, Fabrizio; Miranda, Rick; Zagier, Don; Bombieri, Enrico Appendix to “Polynomial-lemniscates, trees and braids”, Topology, Volume 30 (1991) no. 4, pp. 623-640 | DOI

[9] Catanese, Fabrizio; Paluszny, Marco Polynomial-lemniscates, trees and braids, Topology, Volume 30 (1991) no. 4, pp. 623-640 | DOI | MR | Zbl

[10] Dumitriu, Ioana; Edelman, Alan Matrix models for beta ensembles, J. Math. Phys., Volume 43 (2002) no. 11, pp. 5830-5847 | DOI | MR | Zbl

[11] Edelsbrunner, Herbert; Harer, John Computational topology: an introduction, American Mathematical Society, 2010 | Zbl

[12] Flajolet, Philippe; Sedgewick, Robert Analytic combinatorics, Cambridge University Press, 2009 | Zbl

[13] Frolova, Anastasia; Khavinson, Dmitry; Vasil’ev, Alexander Polynomial lemniscates and their fingerprints: from geometry to topology, Complex analysis and dynamical systems (Trends in Mathematics), Birkhäuser/Springer, 2018, pp. 103-128 | DOI | Zbl

[14] Fyodorov, Yan V; Lerario, Antonio; Lundberg, Erik On the number of connected components of random algebraic hypersurfaces, J. Geom. Phys., Volume 95 (2015), pp. 1-20 | DOI | MR | Zbl

[15] Gayet, Damien; Welschinger, Jean-Yves Exponential rarefaction of real curves with many components, Publ. Math., Inst. Hautes Étud. Sci., Volume 113 (2011) no. 1, pp. 69-96 | DOI | Numdam | MR | Zbl

[16] Gayet, Damien; Welschinger, Jean-Yves Lower estimates for the expected Betti numbers of random real hypersurfaces, J. Lond. Math. Soc., Volume 90 (2014) no. 1, pp. 105-120 | DOI | MR | Zbl

[17] Gayet, Damien; Welschinger, Jean-Yves Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. Eur. Math. Soc., Volume 18 (2016) no. 4, pp. 733-772 | DOI | MR | Zbl

[18] Hanin, Boris Correlations and pairing between zeros and critical points of Gaussian random polynomials, Int. Math. Res. Not., Volume 2015 (2015) no. 2, pp. 381-421 | DOI | MR | Zbl

[19] Hanin, Boris Pairing of zeros and critical points for random meromorphic functions on Riemann surfaces, Math. Res. Lett., Volume 22 (2015) no. 1, pp. 111-140 | DOI | MR | Zbl

[20] Hanin, Boris Pairing of zeros and critical points for random polynomials, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 53 (2017) no. 3, pp. 1498-1511 | MR | Zbl

[21] Lerario, Antonio; Lundberg, Erik Statistics on Hilbert’s 16th problem, Int. Math. Res. Not. (2015) no. 12, pp. 4293-4321 | MR | Zbl

[22] Lerario, Antonio; Lundberg, Erik On the geometry of random lemniscates, Proc. Lond. Math. Soc., Volume 113 (2016) no. 5, pp. 649-673 | DOI | MR | Zbl

[23] Lubinsky, Doron S.; Pritsker, Igor E.; Xie, Xiaoju Expected number of real zeros for random linear combinations of orthogonal polynomials, Proc. Am. Math. Soc., Volume 144 (2016) no. 4, pp. 1631-1642 | DOI | MR | Zbl

[24] Lundberg, Erik; Ramachandran, Koushik The arc length and topology of a random lemniscate, J. Lond. Math. Soc., Volume 96 (2017) no. 3, pp. 621-641 | DOI | MR | Zbl

[25] Nazarov, Fedor; Sodin, Mikhail On the number of nodal domains of random spherical harmonics, Am. J. Math., Volume 131 (2009) no. 5, pp. 1337-1357 | DOI | MR | Zbl

[26] Nazarov, Fedor; Sodin, Mikhail Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, J. Math. Phys. Anal. Geom., Volume 12 (2016) no. 3, pp. 205-278 | MR | Zbl

[27] Nicolaescu, Liviu I Counting Morse functions on the 2-sphere, Compos. Math., Volume 144 (2008) no. 5, pp. 1081-1106 | DOI | MR | Zbl

[28] O’Rourke, Sean; Williams, Noah Pairing between zeros and critical points of random polynomials with independent roots, Trans. Am. Math. Soc., Volume 371 (2019) no. 4, pp. 2343-2381 | DOI | MR | Zbl

[29] Sarnak, Peter; Wigman, Igor Topologies of Nodal Sets of Random Band-Limited Functions, Commun. Pure Appl. Math., Volume 72 (2019) no. 2, pp. 275-342 | DOI | MR | Zbl

[30] Titchmarsh, Edward C The theory of functions, Oxford University Press, 1939

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