TropicalHomotopies.jl

This project is a proof of concept a general framework for solving polynomial systems numerically, based on homotopies constructed using tropical geometry. It uses functionality from OSCAR and HomotopyContinuation.j.

The underlying ideas are to be explained in a forthcoming manuscript titled A tropical method for solving parametrized polynomial systems by Paul Helminck, Oskar Henriksson, and Yue Ren.

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.