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 $\{X^n_d\}_d$ in $\R
\PP ^n$ such that the $i$-th Betti numbers $b_i(\R X^n_d)$ are asymptotically strictly
greater than the $(i,n-1-i)$-th Hodge numbers $h^{i,n-1-i}(\C X^n _d)$.
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.
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