On the total curvature of tropical hypersurfaces

TL;DR

This paper investigates the curvatures of tropical and real tropical hypersurfaces, establishing their equivalence in non-singular cases and deriving sharp inequalities relating their curvatures.

Contribution

It introduces the concepts of total curvature and polyhedral total curvature for tropical hypersurfaces and proves their equivalence in non-singular cases, along with universal inequalities.

Findings

Total curvature and polyhedral total curvature coincide for non-singular tropical hypersurfaces.

Universal inequalities relate the curvatures of tropical and algebraic hypersurfaces.

Inequalities are sharp in the non-singular case.

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 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: - 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 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.