# Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Damien Gayet; Jean-Yves Welschinger

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 4, page 733-772
- ISSN: 1435-9855

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topGayet, Damien, and Welschinger, Jean-Yves. "Betti numbers of random real hypersurfaces and determinants of random symmetric matrices." Journal of the European Mathematical Society 018.4 (2016): 733-772. <http://eudml.org/doc/277493>.

@article{Gayet2016,

abstract = {We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the random real hypersurfaces.},

author = {Gayet, Damien, Welschinger, Jean-Yves},

journal = {Journal of the European Mathematical Society},

keywords = {real projective manifold; ample line bundle; random matrix; random polynomial; real projective manifold; ample line bundle; random matrix; random polynomial},

language = {eng},

number = {4},

pages = {733-772},

publisher = {European Mathematical Society Publishing House},

title = {Betti numbers of random real hypersurfaces and determinants of random symmetric matrices},

url = {http://eudml.org/doc/277493},

volume = {018},

year = {2016},

}

TY - JOUR

AU - Gayet, Damien

AU - Welschinger, Jean-Yves

TI - Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

JO - Journal of the European Mathematical Society

PY - 2016

PB - European Mathematical Society Publishing House

VL - 018

IS - 4

SP - 733

EP - 772

AB - We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the random real hypersurfaces.

LA - eng

KW - real projective manifold; ample line bundle; random matrix; random polynomial; real projective manifold; ample line bundle; random matrix; random polynomial

UR - http://eudml.org/doc/277493

ER -

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