# A Special Homotopy Continuation Method For A Class of Polynomial Systems

## TL;DR

This paper introduces a novel homotopy continuation method combining polyhedral and linear product techniques to efficiently compute solutions of specific polynomial systems, with applications in real algebraic geometry.

## Contribution

It proposes a new homotopy method with a root bound between total degree and mixed volume, and implements it as the LPH program for improved efficiency.

## Key findings

- The method computes all isolated solutions efficiently.
- The root number bound is easily calculated and tighter than traditional bounds.
- The algorithm can find witness points on real variety components.

## Abstract

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety.