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Comptes Rendus Mathématique
Volume 345, n° 9
pages 507-512 (novembre 2007)
Doi : 10.1016/j.crma.2007.10.013
Received : 5 September 2007 ;  accepted : 24 September 2007
Average Euler characteristic of random real algebraic varieties
La caractéristique dʼEuler moyenne des variétés algébriques réelles aléatoires
 

Peter Bürgisser 1
Institute of Mathematics, University of Paderborn, 33095 Paderborn, Germany 

Abstract

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. To cite this article: P. Bürgisser, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.
Résumé

Dans cet article, nous déterminons lʼespérance du polynôme de courbure dʼune 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 dʼEuler de telles variétés projectives réelles aléatoires. Ces résultats généralisent considérablement la connaissance du nombre de zéros, du volume, et de la caractéristique dʼEuler, des ensembles de zéro des systèmes de polynômes aléatoires. Pour citer cet article : P. Bürgisser, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.
1  Partially supported by DFG grant BU 1371/2-1 and Paderborn Institute for Scientific Computation.


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