Asymptotically maximal families of hypersurfaces in toric varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $\Delta$ there are families of hypersurfaces with the Newton polytopes $(\lambda\Delta)_{\lambda \in \mathbb{N}}$ that are asymptotically maximal when $\lambda$ tends to infinity. We also show that these results generalize to complete intersections.Comment: 18 pages, 1 figur

On total reality of meromorphic functions

We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the image.Comment: 13 page

A Non-Algebraic Patchwork

Itenberg and Shustin's pseudoholomorphic curve patchworking is in principle more flexible than Viro's original algebraic one. It was natural to wonder if the former method allows one to construct non-algebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in $\Sigma_n$ whose position with respect to the pencil of lines cannot be realised by any homologous real algebraic curve.Comment: 6 pages, 1 figur