Rigid homotopies for sampling from algebraic varieties: a Waring structure complexity model

TL;DR

This paper introduces new complexity bounds for rigid homotopies in solving polynomial systems with Waring representations and presents initial computational experiments demonstrating their practical application.

Contribution

It establishes a novel complexity result for rigid homotopies tailored to polynomial systems with Waring structure and provides the first experimental validation.

Findings

Proved a new complexity bound for rigid homotopies with Waring representations.

Developed a preliminary implementation for computational experiments.

Demonstrated the practical feasibility of rigid homotopies in polynomial system solving.

Abstract

Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale's 17th problem, establishing average-case polynomial-time algorithms for computing approximate solutions of polynomial systems via homotopy continuation. Recent improvements in complexity bounds for these algorithms led to the development of rigid homotopy methods. In this article, we prove a new complexity result for rigid homotopies for polynomial systems with Waring representations of prescribed length. In addition, we provide the first computational experiments for rigid homotopies using a preliminary implementation.