Numeric certified algorithm for the topology of resultant and discriminant curves

TL;DR

This paper introduces a numerically certified algorithm to determine the topology of resultant and discriminant curves, effectively handling singularities with practical criteria and outperforming existing symbolic methods.

Contribution

It presents a novel numerical approach using subresultants and interval arithmetic to certify singularities and local topology of algebraic curves, addressing limitations of prior algorithms.

Findings

Algorithm successfully certifies singularities in complex curves.

Outperforms symbolic and homotopic methods in efficiency.

Provides practical criteria for topological certification in singular regions.

Abstract

Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$ (resp. $P(x,y,z)=\frac{\partial P}{\partial z}(x,y,z)=0$), and generically its singularities are nodes (resp. nodes and ordinary cusps). State-of-the-art numerical algorithms compute the topology of smooth curves but usually fail to certify the topology of singular ones. The main challenge is to find practical numerical criteria that guarantee the existence and the uniqueness of a singularity inside a given box $B$, while ensuring that $B$ does not contain any closed loop of $\mathcal{C}$. We solve this problem by first providing a square deflation system, based on subresultants, that can be used to certify numerically whether $B$ contains a unique singularity $p$ or not. Then we introduce a numeric adaptive separation criterion based on interval arithmetic to ensure that the topology of $\mathcal C$ in $B$ is homeomorphic to the local topology at $p$. Our algorithms are implemented and experiments show their efficiency compared to state-of-the-art symbolic or homotopic methods.