Maximal and typical topology of real polynomial singularities
Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 589-626.

We study the structure of polynomial singularities given by semialgebraic conditions on the jet of maps from the sphere to Euclidean space. We prove upper and lower bounds for the homological complexity of these singularities. The upper bound is proved using a semialgebraic version of stratified Morse Theory for jets. For the lower bound, we prove a general result stating that small continuous perturbations of C 1 manifolds can only enrich their topology. In the case of random maps, we provide asymptotic estimates for the expectation of the homological complexity, generalizing classical results of Edelman–Kostlan–Shub–Smale.

Nous étudions la structure des singularités polynomiales données par des conditions semi-algébriques sur le jet de fonctions de la sphère à l’espace euclidien. Nous prouvons des bornes supérieure et inférieure pour la complexité homologique de ces singularités. La limite supérieure est prouvée en utilisant une version semi-gébrique de la théorie de Morse stratifiée pour les jets. Pour la borne inférieure, nous prouvons un résultat général indiquant que de petites perturbations continues des variétés C 1 ne peuvent qu’enrichir leur topologie. Dans le cas des fonctions aléatoires, nous fournissons des estimations asymptotiques de l’espérance de la complexité homologique, généralisant des résultats classiques d’Edelman–Kostlan–Shub–Smale.

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DOI: 10.5802/aif.3603
Classification: 14P10, 14P25, 57N65
Keywords: Real Algebraic Geometry, Singularity Theory, Stratified Morse Theory, Random Geometry.
Mot clés : Géométrie algébrique réelle, théorie de la singularité, théorie de Morse stratifiée, géométrie aléatoire

Lerario, Antonio 1; Stecconi, Michele 2

1 SISSA, via Bonomea 265, Trieste (Italy)
2 Department of Mathematics Université du Luxembourg Maison du Nombre 6, Avenue de la Fonte L-4364 Esch-sur-Alzette (Luxembourg)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Lerario, Antonio; Stecconi, Michele. Maximal and typical topology of real polynomial singularities. Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 589-626. doi : 10.5802/aif.3603. https://aif.centre-mersenne.org/articles/10.5802/aif.3603/

[1] Billingsley, Patrick Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, x+277 pages | DOI | MR | Zbl

[2] Breiding, Paul; Keneshlou, Hanieh; Lerario, Antonio Quantitative singularity theory for random polynomials, Int. Math. Res. Not. IMRN (2022) no. 8, pp. 5685-5719 | DOI | MR | Zbl

[3] Bürgisser, Peter Average Euler characteristic of random real algebraic varieties, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 9, pp. 507-512 | DOI | Numdam | MR | Zbl

[4] Cartwright, Dustin; Sturmfels, Bernd The number of eigenvalues of a tensor, Linear Algebra Appl., Volume 438 (2013) no. 2, pp. 942-952 | DOI | MR | Zbl

[5] Delbruck, M. Knotting problems in biology, Plant Genome Data and Information Center collection on computational molecular biology and genetics (1961), 9 pages

[6] Diatta, Daouda Niang; Lerario, Antonio Low-degree approximation of random polynomials, Found. Comput. Math., Volume 22 (2022) no. 1, pp. 77-97 | DOI | MR | Zbl

[7] Edelman, Alan; Kostlan, Eric How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.), Volume 32 (1995) no. 1, pp. 1-37 | DOI | MR | Zbl

[8] Eliashberg, Y.; Mishachev, N. Introduction to the h-principle, Graduate Studies in Mathematics, 48, American Mathematical Society, Providence, RI, 2002, xviii+206 pages | DOI | MR | Zbl

[9] Frisch, Harry L.; Wasserman, Edel Chemical topology, Journal of the American Chemical Society, Volume 83 (1961) no. 18, p. 3789–3795

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

[11] Gayet, Damien; Welschinger, Jean-Yves Lower estimates for the expected Betti numbers of random real hypersurfaces, J. Lond. Math. Soc. (2), Volume 90 (2014) no. 1, pp. 105-120 | DOI | MR | Zbl

[12] Gayet, Damien; Welschinger, Jean-Yves Expected topology of random real algebraic submanifolds, J. Inst. Math. Jussieu, Volume 14 (2015) no. 4, pp. 673-702 | DOI | MR | Zbl

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

[14] Goresky, Mark; MacPherson, Robert Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 14, Springer-Verlag, Berlin, 1988, xiv+272 pages | DOI | MR | Zbl

[15] Hirsch, Morris W. Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994, x+222 pages (Corrected reprint of the 1976 original) | MR

[16] Itô, Kiyosi; Nisio, Makiko On the convergence of sums of independent Banach space valued random variables, Osaka Math. J., Volume 5 (1968), pp. 35-48 http://projecteuclid.org/euclid.ojm/1200692040 | MR | Zbl

[17] Kostlan, E. On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, New York, 1993, pp. 419-431 | DOI | MR | Zbl

[18] Kozhasov, Khazhgali On fully real eigenconfigurations of tensors, SIAM J. Appl. Algebra Geom., Volume 2 (2018) no. 2, pp. 339-347 | DOI | MR | Zbl

[19] Lerario, A. Complexity of intersections of real quadrics and topology of symmetric determinantal varieties, J. Eur. Math. Soc. (JEMS), Volume 18 (2016) no. 2, pp. 353-379 | DOI | MR | Zbl

[20] Lerario, Antonio; Stecconi, Michele Differential Topology of Gaussian Random Fields (2019) (https://arxiv.org/abs/1902.03805)

[21] Milnor, J. On the Betti numbers of real varieties, Proc. Amer. Math. Soc., Volume 15 (1964), pp. 275-280 | DOI | MR | Zbl

[22] Nazarov, F.; Sodin, M. Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, Zh. Mat. Fiz. Anal. Geom., Volume 12 (2016) no. 3, pp. 205-278 | DOI | MR | Zbl

[23] Podkorytov, S. S. On the Euler characteristic of a random algebraic hypersurface, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 252 (1998) no. Geom. i Topol. 3, p. 224-230, 252–253 | DOI | MR

[24] Sarnak, Peter; Wigman, Igor Topologies of nodal sets of random band limited functions, Advances in the theory of automorphic forms and their L-functions (Contemp. Math.), Volume 664, American Mathematical Society, Providence, RI, 2016, pp. 351-365 | DOI | MR | Zbl

[25] Shub, M.; Smale, S. Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) (Progr. Math.), Volume 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267-285 | DOI | MR | Zbl

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