Tropical Implicitization Revisited

TL;DR

This paper introduces a new implementation for tropical implicitization in Oscar.jl, efficiently computing Newton polytopes for hypersurfaces and extending the approach to higher codimension via Chow forms, with open questions posed.

Contribution

It provides a novel implementation of tropical implicitization in Oscar.jl and extends the methodology to higher codimension cases using Chow forms.

Findings

Successfully solves challenging instances of tropical implicitization

Can be applied to classical implicitization problems

Extends the approach to higher codimension via Chow forms

Abstract

Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its coefficients. We present a new implementation of this procedure in Oscar.jl. It solves challenging instances, and can be used for classical implicitization as well. We also develop implicitization in higher codimension via Chow forms, and we pose several open questions.