TropicalHomotopies.jl

This project is a proof of concept for a general framework for solving polynomial systems through homotopies constructed using tropical geometry, developed in the paper A tropical method for solving parametrized polynomial systems by Paul Helminck, Oskar Henriksson, and Yue Ren. The code uses functionality from OSCAR and HomotopyContinuation.j.

Structure of the repository

The repository consists of three main parts:

• The examples directory contains explicit computations used for the examples given throughout the paper.
• The src directory contains an implementation of the main algorithm from the paper in the case of vertically parametrized systems.
• The case_studies contains code for the examples from chemical reaction network theory and graph rigidity theory discussed at the end of the paper, that uses functionality from src.

Warning

The path-tracking portion of the code is still experimental, and sometimes displays numerical instabilities.

Examples

See the notebook case_studies/wnt_pathway.ipynb.

Installation

The easiest way to get started using the code in src is to manually download all files, and then run

include("PATH/TO/src/main.jl")

where PATH/TO/src/main.jl is replaced by the actual path to the file src/main.jl.

Note

Parts of the code require the latest development version of version 1.2 of OSCAR (09/09/2024), and version 2.11.0 of HomotopyContinuation.jl.