A Companion Curve Tracing Method for Rank-deficient Polynomial Systems

TL;DR

This paper introduces a novel method for tracing real algebraic curves defined by polynomial systems with rank-deficient Jacobians, combining regularization, witness point computation, and numerical continuation for accurate curve approximation.

Contribution

It presents a new approach that integrates regularization, witness point refinement, and trajectory-based numerical continuation to trace implicit algebraic curves with rank-deficient systems.

Findings

Successfully computes witness points for each connected component.

Provides a sufficient condition for testing emptiness of the curve.

Demonstrates effectiveness through illustrative examples.

Abstract

We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve $f^{-1}(0)$, it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of $f^{-1}(0)$. We also analyze the convergence rate and carry out an error analysis for refining the witness points. The witness points are obtained by computing the minimum distance of a random point to a smooth manifold embedding the curve while at the same time penalizing the residual of $f$ at the local minima. To trace the curve starting from these witness points, we prove that if one drags the random point along a trajectory inside a tubular neighborhood of the embedded manifold of the curve, the projection of the trajectory on the manifold is unique and can be computed by numerical continuation. We then show how to choose such a trajectory to approximate the curve by computing eigenvectors of certain matrices. Effectiveness of the method is illustrated by examples.