Comptes Rendus
Probability Theory
Average Euler characteristic of random real algebraic varieties
Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 507-512.

We determine the expected curvature polynomial of real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, which is invariant under the orthogonal group. In particular, the expected Euler characteristic of such random real projective varieties is found. This considerably extends previously known results on the number of roots, the volume, and the Euler characteristic of the real solution set of random polynomial equations.

Dans cet article, nous déterminons l'espérance du polynôme de courbure d'une variété projective réelle qui est donnée comme ensemble de zéros de polynômes aléatoires, avec une distribution gaussienne qui est invariante par le groupe orthogonal. En particulier, nous explicitons la caractéristique d'Euler de telles variétés projectives réelles aléatoires. Ces résultats généralisent considérablement la connaissance du nombre de zéros, du volume, et de la caractéristique d'Euler, des ensembles de zéro des systèmes de polynômes aléatoires.

Published online:
DOI: 10.1016/j.crma.2007.10.013

Peter Bürgisser 1

1 Institute of Mathematics, University of Paderborn, 33095 Paderborn, Germany
     author = {Peter B\"urgisser},
     title = {Average {Euler} characteristic of random real algebraic varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {507--512},
     publisher = {Elsevier},
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     number = {9},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.013},
     language = {en},
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DO  - 10.1016/j.crma.2007.10.013
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Peter Bürgisser. Average Euler characteristic of random real algebraic varieties. Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 507-512. doi : 10.1016/j.crma.2007.10.013.

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