On the total curvature of tropical hypersurfaces
Benoît Bertrand, Lucía López de Medrano, Jean-Jacques Risler · Contemporary mathematics - American Mathematical Society · 2013
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 V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part R V \mathbb {R} V which is a polyhedral complex. We define the total curvature of V V (resp. R V \mathbb {R}V ) by using the total curvature of Amoebas and passing to the limit. We also define the “polyhedral total curvature” of the real part R V \mathbb {R} V of a generic tropical hypersurface. The main results we prove about these notions are the following: The fact that the total curvature and the polyhedral total curvature coincide for real non-singular tropical hypersurfaces. A universal inequality between the total curvatures of V V and R V \mathbb {R} V 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.