On the Number of Connected Components of T-Hypersurfaces

A T-hypersurface is a combinatorial hypersurface of the real locus of a projective toric variety $Y$. It is constructed from a primitive triangulation $K$ of a moment polytope $P$ of $Y$ and a $0$-cochain $\varepsilon$ on $K$ with coefficients in the field with two elements $\mathbb{F}_2$, called a sign distribution. O. Viro showed that when $K$ is convex the T-hypersurface is ambiantly isotopic to a real algebraic hypersurface of $Y$. A. Renaudineau and K. Shaw gave upper bounds on the Betti numbers of T-hypersurfaces in terms of the Hodge numbers of a generic section of the ample line bundle $L$ associated with the moment polytope. In particular, the number of connected components of a T-hypersurface cannot exceed the geometric genus of a generic section of $L$ plus one. In this article we investigate whether this upper bound is attainable. We are able to characterise the couples $(K;\varepsilon)$ leading to T-hypersurfaces realising the Renaudineau-Shaw upper bound on the number of connected components. This theorem generalises B. Haas' theorem for T-curves. In contrast with this results we find that the upper bound is not always attainable on every primitive triangulations. For some of those on which it is not attainable we provide a sharper upper bound. Finally we use our characterisation to show that there always exists a triangulation and a sign distribution on the standard simplex that reach the Renaudineau-Shaw upper bound. We also study the growth of the expected number of connected components of a T-hypersurface as we dilate the moment polytope by $d$ (i.e. we tensorise the line bundle $d$-times with itself) and show that it is always of the order of $d^n$ where $n$ is the dimension of $P$.