Upper estimates for the expected Betti numbers of random subcomplexes (Q6592026)

What is the expected number of real roots of a real polynomial in one variable? A collection of results have been obtained [\textit{D. Gayet} and \textit{J.-Y. Welschinger}, Asian J. Math. 21, No. 5, 811--840 (2017; Zbl 1394.58011)], especially about the asymptotic topology of such submanifolds when their degrees grow to \(+\infty\), but even the convergence of the rescaled asymptotic Betti numbers is unknown in positive degrees (see [\textit{F. Nazarov} and \textit{M. Sodin}, J. Math. Phys. Anal. Geom. 12, No. 3, 205--278 (2016; Zbl 1358.60057)] for degree 0). Upper and lower estimates for the latter are known through [\textit{D. Gayet} and \textit{J.-Y. Welschinger}, J. Eur. Math. Soc. (JEMS) 18, No. 4, 733--772 (2016; Zbl 1408.14187)] in terms of random matrices for the upper ones. Nodal sets provide another domain where similar questions of random topology are under study by now [\textit{D. Gayet} and \textit{J.-Y. Welschinger}, op. cit.]. What is the expected topology of a random nodal set, that is of the vanishing locus of a random linear combination of the eigenvectors of a Laplace-Beltrami operator on a Riemannian manifold? A special class of real projective manifolds are toric ones, where many features of algebraic geometry can be computed in a combinatorial way on the moment polytope of the manifold. The authors of the present paper introduced a model of random topology, a discrete geometric one, where the manifold or triangulated moment polytope is replaced by a finite simplicial complex, keeping the construction of hypersurfaces which holds in this generality and raising the question of their expected topology. It turns out that it is possible to prove the convergence of the asymptotic Betti numbers in this model and the aim of the paper is to estimate from above their limit. The authors observe that every zero-dimensional simplicial cochain defines a canonical filtration of a finite simplicial complex and deduce upper estimates for the expected Betti numbers of codimension one random subcomplexes in its support.