Computing the Real Isolated Points of an Algebraic Hypersurface

TL;DR

This paper presents a probabilistic algorithm for efficiently computing real isolated points of algebraic hypersurfaces, aiding in the analysis of rigidity in material design.

Contribution

It introduces a novel probabilistic method combining critical point computation and roadmaps to find isolated points in real algebraic sets, with practical efficiency improvements.

Findings

Algorithm handles high-degree hypersurfaces effectively.

Complexity is $(nd)^{O(n ext{log}(n))}$, enabling practical solutions.

Outperforms existing methods on challenging instances.

Abstract

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations of critical points as well as roadmaps for answering connectivity queries in real algebraic sets. This leads to a probabilistic algorithm of complexity $(nd)^{O(n\log(n))}$ for computing the real isolated points of real algebraic hypersurfaces of degree $d$. It allows us to solve in practice instances which are out of reach of the state-of-the-art.