Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers

In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension $n$ from families of such hypersurfaces in ambient dimensions $k=1,\ldots,n-1$. The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of the families used as ingredients. The algorithm is based on Viro's Patchwork and inspired by I. Itenberg's and O. Viro's construction of asymptotically maximal families in arbitrary dimension. Using it, we prove that for any $n$ and $i=0,\ldots,n-1$, there is a family of asymptotically maximal real projective algebraic hypersurfaces $\{Y^n_d\}_d$ in $\mathbb{R} \mathbb{P} ^n$ (where $d$ denotes the degree of $Y^n_d$) such that the $i$-th Betti numbers $b_i(\mathbb{R} Y^n_d)$ are asymptotically strictly greater than the $(i,n-1-i)$-th Hodge numbers $h^{i,n-1-i}(\mathbb{C} Y^n_d)$. We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree $d$) Betti numbers that are asymptotically (in the ambient dimension $n$) very large.