Reducing smooth functions 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 of the functions by some coordinate changes in some neighborhoods of these points. A function 𝑓(𝑥, 𝑦) has a singularity of the type 𝐴𝑘, 𝐸6, or 𝐸8 at its critical point if, in some local coordinate system centered at this point, the Taylor series of the function has the form 𝑥2 +𝑦𝑘+1 +𝑅2,𝑘+1, 𝑥3 +𝑦4 +𝑅3,4, 𝑥3 +𝑦5 +𝑅3,5 respectively, where 𝑅𝑚,𝑛 stands for a sum of higher order terms, i.e., 𝑅𝑚,𝑛 = Σ︀𝑎𝑖𝑗𝑥𝑖𝑦𝑗 where 𝑖/𝑚 + 𝑗/𝑛 > 1. In according to a result by V. I. Arnold (1972), these singularities are simple and can be reduced to the canonical form with 𝑅𝑚,𝑛 = 0 by a smooth coordinate change.
For the singularity types 𝐴𝑘, 𝐸6, and 𝐸8, we explicitly construct such a coordinate change and estimate from below (in terms of 𝐶𝑟-norm of the function, where 𝑟 = 𝑘 + 3, 7, and 8 respectively) the maximal radius of a neighborhood in which the coordinate change is defined.
Our coordinate change provides a “uniform” reduction to the canonical form in the sense that the radius of the neighborhood and the coordinate change we constructed in it (as well as all partial derivatives of the coordinate change) continuously depend on the function 𝑓 and its partial derivatives.