On the number of connected components of random algebraic hypersurfaces

We study the expectation of the number of components b0(X)b0(X) of a random algebraic hypersurface XX defined by the zero set in projective space RPn of a random homogeneous polynomial ff of degree dd. Specifically, we consider invariant ensembles, that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables.Fixing nn, under some rescaling assumptions on the family of ensembles (as d→∞d→∞), we prove that Eb0(X)Eb0(X) has the same order of growth as [Eb0(X∩RP1)]n. This relates the average number of components of XX to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial f|RP1.The proof requires an upper bound for Eb0(X)Eb0(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→∞n→∞) of the leading coefficient (in dd) of Eb0(X)Eb0(X).