A search for an optimal start system for numerical homotopy continuation

TL;DR

This paper investigates methods to identify optimal start systems for numerical homotopy continuation, demonstrating that better average complexity solutions can be found compared to traditional systems, through experimental approaches.

Contribution

The authors develop and utilize a certified homotopy tracking algorithm to empirically search for start systems that reduce the average complexity of root-finding in polynomial systems.

Findings

Identified start systems with lower average complexity than standard ones

Experimental evidence supports the feasibility of improving homotopy continuation efficiency

Showed that optimal start systems are attainable despite the problem's hardness

Abstract

We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start systems is a hard problem, our experiments show that it is possible to find start systems that deliver better average complexity than the ones that are commonly used in the existing homotopy continuation software.