Four families of maximal real algebraic hypersurfaces in ${\mathbb{R}\mathbb{P}}^4$

In this paper, we present four families of maximal real algebraic hypersurfaces of even degree in ${\mathbb{R}\mathbb{P}}^4$ constructed using O. Viro’s combinatorial patchworking method. We compare the Euler characteristic of the real part and the signature of the complex part of double coverings of ${\mathbb{C}\mathbb{P}}^4$ ramified over the complex part of the constructed real algebraic hypersurfaces. We prove that these invariants are not necessarily equal and can even be asymptotically different.