Constructive proof of extended Kapranov theorem

TL;DR

This paper provides a new constructive proof of Kapranov's Theorem, a key result in tropical geometry, which also characterizes principal terms of points in hypersurfaces within algebraic tori.

Contribution

It introduces a constructive proof of Kapranov's Theorem, enhancing understanding of tropical geometry and hypersurface principal terms.

Findings

Constructive proof of Kapranov's Theorem presented.

Characterization of principal terms in hypersurfaces in algebraic tori.

Enhanced methods for tropical geometry analysis.

Abstract

Kapranov Theorem is a well known generalization of Newton-Puiseux theorem for the case of several variables. This theorem is stated mainly in the context of tropical geometry. We present a new, constructive proof, that also characterizes the possible principal terms of points in a hypersurface contained in the algebraic torus $(K^*)^n$.