Reducing smooth functions to normal forms near critical points

TL;DR

This paper develops methods to simplify smooth functions near critical points on 2-manifolds into canonical forms, explicitly constructing coordinate changes for certain singularity types and estimating neighborhood sizes.

Contribution

It provides explicit coordinate transformations for singularity types E6, E8, and An, and estimates the neighborhoods where these reductions hold.

Findings

Explicit coordinate changes for E6, E8, and An singularities.

Lower bounds for neighborhood sizes in terms of C^r-norms.

Method for uniform reduction to normal forms near critical points.

Abstract

The paper is devoted to "uniform" reduction of smooth functions on 2-manifolds to canonical form near critical points by some coordinate changes in some neighbourhoods of these points. For singularity types $E_6,E_8$ and $A_n$, we explicitly construct such coordinate changes and estimate from below (in terms of $C^r$-norm of the function) the radius of a required neighbourhood.