Computing complex and real tropical curves using monodromy

TL;DR

This paper introduces algorithms leveraging homotopy continuation, monodromy loops, and Cauchy integrals to compute complex and real tropical curves from polynomial equations, implemented in numerical algebraic geometry software.

Contribution

It presents novel algorithms for computing tropical curves using homotopy and monodromy, with implementation in Bertini for practical computation.

Findings

Algorithms successfully compute tropical curves from polynomials.

Implementation demonstrates practical effectiveness on examples.

Provides new tools for tropical geometry analysis.

Abstract

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.