Generic distance-squared mappings on plane curves

TL;DR

This paper investigates the stability of compositions of plane curves with generic distance-squared mappings, a key concept in singularity theory and differential geometry, revealing their structural properties.

Contribution

It provides a comprehensive analysis of the stability of distance-squared mappings composed with plane curves, extending understanding in singularity theory.

Findings

Distance-squared mappings are stable under generic conditions.

Characterization of stable mappings from plane curves.

Insights into the singularities of composed mappings.

Abstract

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of a given plane curve and generic distance-squared mappings on the plane into the plane are investigated from the viewpoint of stability.