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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     journal = {Annales de l'Institut Fourier},
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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.

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