Khovanskii-Rolle continuation for real solutions

TL;DR

This paper introduces a novel continuation algorithm that efficiently finds all nondegenerate real solutions to polynomial systems by tracing real solution curves, avoiding complex solutions and leveraging the fewnomial bound.

Contribution

It presents a new real-focused continuation method that differs from homotopy approaches by following only real paths and not deforming the system.

Findings

Algorithm successfully finds all real solutions without computing complex ones.

Method exploits the fewnomial bound to optimize the number of curves traced.

Approach is effective for systems with sparse polynomial structures.

Abstract

We present a new continuation algorithm to find all nondegenerate real solutions to a system of polynomial equations. Unlike homotopy methods, it is not based on a deformation of the system; instead, it traces real curves connecting the solutions of one system of equations to those of another, eventually leading to the desired real solutions. It also differs from homotopy methods in that it follows only real paths and computes no complex solutions of the original equations. The number of curves traced is bounded by the fewnomial bound for real solutions, and the method takes advantage of any slack in that bound.