On the number of connected components of random algebraic hypersurfaces

JOURNAL OF GEOMETRY AND PHYSICS

卷:95

Article

Fyodorov, Yan V.¹  Lerario, Antonio²  Lundberg, Erik³ 

[1] Univ London, Sch Math Sci, London E14NS, England

[2] Univ Claude Bernard Lyon 1, 43 Blvd 11 Novembre 1918, F-69622 Villeurbanne, France

[3] Florida Atlantic Univ, Dept Math Sci, Boca Raton, FL 33431 USA

关键词: Real algebraic geometry;    Gaussian field;    Harmonic polynomials;    Critical point theory;    Hilbert's sixteenth problem;   

DOI  :  10.1016/j.geomphys.2015.04.006

来源: Elsevier

【 摘 要 】

We study the expectation of the number of components b(0)(X) of a random algebraic hypersurface X defined by the zero set in projective space RPn of a random homogeneous polynomial f of degree d. Specifically, we consider invariant ensembles, that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. Fixing n, under some rescaling assumptions on the family of ensembles (as d -> infinity), we prove that Eb(0) (X) has the same order of growth as [Eb(0) (X boolean AND RP1)](n). This relates the average number of components of X to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial f vertical bar Rp(1). The proof requires an upper bound for Eb(0) (X), which we obtain by counting extrema using Random Matrix Theory methods from Fyodorov (2013), and it also requires a lower bound, which we obtain by a modification of the barrier method from Lerario and Lundberg (2015) and Nazarov and Sodin (2009). We also provide quantitative upper bounds on implied constants; for the real Fubini Study model these estimates provide super-exponential decay (as n ->infinity) of the leading coefficient (in d) of Eb(0) (X). (C) 2015 Elsevier B.V. All rights reserved.

【 授权许可】

Free