Statistics on Hilbert's Sixteenth Problem

Abstract

We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in RP^n defined by a Real Bombieri-Weyl distributed homogeneous polynomial of degree d. We prove that the expectation of the number of connected components of such hypersurface has order d^n, the asymptotic being in d for n fixed. We do not restrict ourselves to the random homogeneous case and we consider more generally random polynomials belonging to a window of eigenspaces of the laplacian on the sphere S^n, proving that the same asymptotic holds. As for the volume, we prove its expectation is of order d. Both these behaviors exhibit expectation of maximal order in light of Milnor's bound and the a priori bound for the volume.

28 pages, extended version

References in corpus (2)

• Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
• Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

Cited by in corpus (2)

• On the number of connected components of random algebraic hypersurfaces
• Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials