Statistics on Hilbert's 16th Problem

Abstract

We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in <f>$\\mathbb {R}\\hbox {P}^n$</f> defined by a homogeneous polynomial <it>f</it> of degree <it>d</it> in the <it>real Fubini–Study</it> ensemble. We prove that for the expectation of the number of connected components, <fd id="RNU069M1"><no>(1)</no>$$\\label{eq1} \\mathbb{E} b_0(Z_{\\mathbb{R}\\hbox{P}^n}(f))=\\Theta(d^n), $$</fd> the asymptotic being in <it>d</it> for <it>n</it> 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 <it>S</it><it>n</it>, proving that the same asymptotic holds. As for the volume properties, we provide an exact formula: <fd id="RNU069M2"><no>(2)</no>$$\\label{eq2} \\mathbb{E}\\,\\hbox{Vol}(Z_{\\mathbb{R}\\hbox{P}^n}(f)) = \\delta^{1/2}\\,\\text{Vol}(S^{n-1}), $$</fd> where <it>δ</it> (which we specify exactly) is asymptotically a constant times <it>d</it>2. Both Equations (<cross-ref type="fd" refid="RNU069M1">1</cross-ref>) and (<cross-ref type="fd" refid="RNU069M2">2</cross-ref>) exhibit expectation of maximal order in light of Milnor's bound <f>$b_0(Z_{\\mathbb {R}\\hbox {P}^n}(f)) = O(d^n)$</f> and the bound <f>$\\hbox {Vol}(Z_{\\mathbb {R}\\hbox {P}^n}(f)) = O(d)$</f>.