Generalized logarithmic Gauss map and its relation to (co)amoebas

TL;DR

This paper introduces a generalized logarithmic Gauss map for complex algebraic varieties of any codimension, analyzes its critical points, and extends known results from hypersurfaces to higher codimension varieties.

Contribution

It defines a new generalized Gauss map for algebraic varieties of any codimension and extends the analysis of critical points of the logarithmic map beyond hypersurfaces.

Findings

Defined the generalized logarithmic Gauss map for varieties of any codimension

Characterized the set of critical points of the logarithmic map on these varieties

Extended Mikhalkin's result to higher codimension cases

Abstract

We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the logarithmic mapping restricted to our variety, and we show an analogous of Mikhalkin's result on the critical points of the logarithmic map restricted to a hypersurfaces.