A new approach to the family of singularities $Re(x+iy)^m$

TL;DR

This paper provides an alternative proof that certain smooth functions with specific Taylor series expansions are diffeomorphically equivalent to the singularity $Re(x+iy)^m$, and demonstrates non-equivalence for a modified function when $m extgreater 4$.

Contribution

It offers a new proof technique for classifying singularities of the form $Re(x+iy)^m$ and identifies conditions under which functions are not diffeomorphically equivalent to these singularities.

Findings

Functions with Taylor series starting with $Re(x+iy)^m$ are diffeomorphically equivalent to $Re(x+iy)^m$.

For $m extgreater 4$, adding a term $C(x^2+y^2)^{m-2}$ changes the equivalence class.

An alternative proof approach for singularity classification is developed.

Abstract

Assume that $m\ge 2$ and let $l$ be a nonnegative integer with $l\ge m-4$. We give an alternative proof of the fact that any smooth function defined locally around $(0,0)\in \mathbb{R}^2$ with the Taylor power series at $(0,0)$ beginning with $$Re(x+iy)^m+0+...+0$$ ($l$ zeros) is diffeomorphically equivalent to $Re(x+iy)^m$ at $(0,0)$. For $m\ge 5$ and $C\ne 0$ we show that the function $$Re(x+iy)^m+C(x^2+y^2)^{m-2}$$ is not diffeomorphically equivalent to $Re(x+iy)^m$ at $(0,0)$.