Betti numbers of random hypersurface arrangements

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