Amoebas, convexity and the volume of integer polytopes

Abstract

To any given Laurent polynomial $f$ on $\mathbf{C}_{*}^{n}$ we associate two natural convex functions $M_f$ and $N_f$ on $\mathbf{R}^n$. We compute the Hessian of $M_f$ and obtain an explicit formula for the volume of the Newton polytope $\Delta_f$. We also establish asymptotic formulas relating our convex functions to coherent triangulations of $\Delta_f$ and to the secondary polytope.