By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=x'^2+y'^2$.


**Question.** Suppose now we consider functions $f$ with Taylor series starting with $(x^2+y^2)^n$. For which $n>1$ one can always find coordinates $(x',y')$ so that $f(x',y')=(x'^2+y'^2)^n$?



Normal form of functions $(x^2+y^2)^n+$ higher terms
The expression $(x^2+y^2)^2 + x^5 + y^5$ cannot be written in the form $(z^2+w^2)^2$ for any smooth functions $z$ and $w$ of $x$ and $y$. (Just look at the Taylor series expansion.) Similarly, $n>1$, the function $(x^2+y^2)^n + x^{2n+1} + y^{2n+1}$ cannot be written in the form $(z^2+w^2)^n$ for any smooth functions $z$ and $w$ of $x$ and $y$.


