Abstract
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 complex. We define the total of V (resp. RV) by using the total 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: - The fact that the total and the total coincide for real non-singular tropical hypersurfaces. - A universal inequality between the total curvatures of V and RV and another between the logarithmic curvatures of the real and complex parts of a real algebraic hypersurface. -The fact that this inequality is sharp in the non-singular case.
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