Euler characteristic of primitive T-hypersurfaces and maximal surfaces

Viro method plays an important role in the study of topology of real algebraic hypersurfaces. The T-primitive hypersurfaces we study here appear as the result of Viro's combinatorial patchworking when one starts with a primitive triangulation. We show that the Euler characteristic of the real part of such a hypersurface of even dimension is equal to the signature of its complex part. We use this result to prove the existence of maximal surfaces in some three-dimensional toric varieties, namely those corresponding Nakajima polytopes. In fact, these results belong to the field of tropical geometry and we explain how they can be understood tropically.