On the total curvature of tropical hypersurfaces

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about these notions are the following: (1) The fact that the total curvature and the polyhedral total curvature coincide for real non-singular tropical hypersurfaces. (2) A universal inequality between the total curvatures of V and RV and anotherbetween the logarithmic curvatures of the real and complex parts of a real algebraic hypersurface. (3) The fact that this inequality is sharp in the non-singular case.