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 ε on K with coefficients in the field with two elements F₂, 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;ε) 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 dⁿ where n is the dimension of P.