[Topologie maximale et typique des singularités polynomiales réelles]
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 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.
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 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.
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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
CC-BY-ND 4.0
@article{AIF_2024__74_2_589_0,
author = {Lerario, Antonio and Stecconi, Michele},
title = {Maximal and typical topology of real polynomial singularities},
journal = {Annales de l'Institut Fourier},
pages = {589--626},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {74},
number = {2},
year = {2024},
doi = {10.5802/aif.3603},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3603/}
}
TY - JOUR AU - Lerario, Antonio AU - Stecconi, Michele TI - Maximal and typical topology of real polynomial singularities JO - Annales de l'Institut Fourier PY - 2024 SP - 589 EP - 626 VL - 74 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3603/ DO - 10.5802/aif.3603 LA - en ID - AIF_2024__74_2_589_0 ER -
%0 Journal Article %A Lerario, Antonio %A Stecconi, Michele %T Maximal and typical topology of real polynomial singularities %J Annales de l'Institut Fourier %D 2024 %P 589-626 %V 74 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3603/ %R 10.5802/aif.3603 %G en %F AIF_2024__74_2_589_0
Lerario, Antonio; Stecconi, Michele. Maximal and typical topology of real polynomial singularities. Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 589-626. doi : 10.5802/aif.3603. https://aif.centre-mersenne.org/articles/10.5802/aif.3603/
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