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 RP n 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:(1)Eb0(Z RP n (f )) = Θ(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 properties we prove that:(2) EVol(Z RP n (f )) = Θ(d). Both equations (1) and (2) exhibit expectation of maximal order in light of Milnor's bound b0(Z RP n (f )) ≤ O(d n ) and the bound Vol(Z RP n (f )) ≤ O(d).

Universal Components of Random Nodal Sets

“…In this section, we follow the method of [7] partially inspired by [3,12] (see also [5,10]) in order to prove Theorem 0.3. If is a smooth closed hypersurface of R n , x ∈ M and s L ∈ U L is given by Proposition 1.10 and vanishes transversally along L in a small ball B(x, RL − 1 m ), then we decompose any random section s ∈ U L as s = as L + σ , where a ∈ R is Gaussian and σ is taken at random in the orthogonal complement of Rs L in U L .…”

“…In [12], Nazarov and Sodin proved that the expected number of components of the vanishing locus of random eigenfunctions with eigenvalue L of the Laplace operator on the round 2-sphere is asymptotic to a constant times L. In the recent [13], they obtain similar results in a more general setting, in particular for all round spheres and flat tori (see also [15]). In [10], Lerario and Lundberg proved, for the Laplace operator on the round n-sphere, the existence of a positive constant c such that E(b 0 ) ≥ c √ L n for large values of L. We got in [6] upper estimates for lim sup L→+∞ L − n m E(b i ) under the same hypotheses as Corollary 0.2, and previously obtained similar upper and lower estimates for the expected Betti numbers or N 's of random real algebraic hypersurfaces of real projective manifolds (see [4,5,7,8]). …”

“…The second section is devoted to the proofs of Theorems 0.1 and 0.3, and of Corollary 0.2. For this purpose we follow the approach used in [7] (see also [5]), which was itself partially inspired by the works [3,12], see also [10]. We begin by estimating the expected local C 1 -norm of elements of U L , see Proposition 2.1, and then compare it with the amount of transversality of s L .…”

Topologies of Nodal Sets of Random Band‐Limited Functions

“…Specifically, they show that the number of connected components of such nodal sets obey an asymptotic law. In [29] we pointed out that these may be applied to ovals of a random real plane curve, and in [21] this is extended to real hypersurfaces in P n . In [15] the barrier technique from [26] is used to show that "all topologies" occur with positive probability in the context of real sections of high tensor powers of a holomorphic line bundle of positive curvature on a real projective manifold.…”

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

“…The second is the (ongoing) investigation of F. Nazarov and M. Sodin [29,30] concerning the topology of the zero set of a random function on a manifold. Finally, we want to mention the recent work of A. Lerario and E. Lundberg [25] on a probabilistic version of Hilbert's 16th problem.…”