Statistics on Hilbert’s Sixteenth Problem

We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in RPn defined by a homogeneous polynomial f of degree d in the real Fubini-Study ensemble. We prove that for the expectation of the number of connected components, Eb(0)(Z(RPn)(f)) = Theta(d(n)), (1) 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 properties, we provide an exact formula: EVol(Z(RPn)(f))= delta(1/2)Vol(Sn-1), (2) where delta (which we specify exactly) is asymptotically a constant times d(2). Both Equations (1) and (2) exhibit expectation of maximal order in light of Milnor's bound b(0)(Z(RPn)(f))= O(d(n)) and the bound Vol(Z(RPn)(f))= O(d).