Clustering polar curves

TL;DR

This paper explores the classification of polar curves of 2-variable function germs into clusters, establishing a topological and Lipschitz correspondence that refines the understanding of their structure.

Contribution

It introduces a bijective correspondence between polar quotient partitions and gradient canyons, enhancing the classification of polar curves in different categories.

Findings

Establishes a topological bijection between polar quotient partitions and polar clusters.

Refines the classification in the Lipschitz category using gradient canyons.

Provides a framework for understanding polar curves through topological and Lipschitz perspectives.

Abstract

This essay builds on the idea of grouping the polar curves of 2-variable function germs into polar clusters. In the topological category, one obtains a bijective correspondence between certain partitions of the polar quotients of two topologically equivalent function germs. We explain how this bijective correspondence may be refined in the Lipschitz category in terms of the associated gradient canyons.