Reducing smooth functions to normal forms near critical points

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 $A_k$, $E_6$, or $E_8$ at its critical point if, in some local coordinate system centered at this point, the Taylor series of the function has the form $x^2+y^{k+1}+R_{2,k+1}$, $x^3+y^4+R_{3,4}$, $x^3+y^5+R_{3,5}$ respectively, where $R_{m,n}$ stands for a sum of higher order terms, i.e., $R_{m,n}=\sum a_{ij}x^iy^j$ where $\frac{i}{m}+\frac{j}{n}>1$. In according to a result by V. I. Arnold (1972), these singularities are simple and can be reduced to the canonical form with $R_{m,n}=0$ by a smooth coordinate change.
For the singularity types $A_k$, $E_6$, and $E_8$, we explicitly construct such a coordinate change and estimate from below (in terms of $C^r$-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.