# Beyond Polyhedral Homotopies

## TL;DR

This paper introduces a novel algorithmic framework combining tropical geometry and homotopy continuation to solve polynomial systems, extending the capabilities of existing polyhedral homotopies.

## Contribution

It generalizes polyhedral homotopies by incorporating tropical geometry and linear subspace structures for solving polynomial equations.

## Key findings

- Successfully extends polyhedral homotopies
- Demonstrates effectiveness on complex polynomial systems
- Provides a new computational approach for algebraic geometry

## Abstract

We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring. This approach generalizes the polyhedral homotopies by Huber and Sturmfels.