Betti Numbers of Random Hypersurface Arrangements

TL;DR

This paper analyzes the expected Betti numbers of random hypersurface arrangements in real projective space, providing asymptotic estimates and connecting the problem to random graph models, especially for quadratic hypersurfaces.

Contribution

It introduces a random spectral sequence approach to estimate Betti numbers and establishes a novel link between hypersurface arrangements and random geometric graphs.

Findings

Expected number of connected components grows linearly with the number of polynomials.

For quadratic hypersurfaces, the expected number of components is sublinear.

A new bound on the Betti numbers of random quadratic arrangements is derived.

Abstract

We study the expected behavior of the Betti numbers of arrangements of the zeros of random (distributed according to the Kostlan distribution) polynomials in $\mathbb{R}\mathrm{P}^n$. Using a random spectral sequence, we prove an asymptotically exact estimate on the expected number of connected components in the complement of $s$ such hypersurfaces in $\mathbb{R}\mathrm{P}^n$. We also investigate the same problem in the case where the hypersurfaces are defined by random quadratic polynomials. In this case, we establish a connection between the Betti numbers of such arrangements with the expected behavior of a certain model of a randomly defined geometric graph. While our general result implies that the average zeroth Betti number of the union of random hypersurface arrangements is bounded from above by a function that grows linearly in the number of polynomials in the arrangement, using the connection with random graphs, we show an upper bound on the expected zeroth Betti number of random quadrics arrangements that is sublinear in the number of polynomials in the arrangement. This bound is a consequence of a general result on the expected number of connected components in our random graph model which could be of independent interest.