$f(u,v)=uv+u^2v$ can be written as $u^2+v^2$ by the change of coordinate in some neighborhood of $(0,0)$.

Question

Proposition: Let $f:\mathbb{C}^2\to \mathbb{C}$ such that $f(0,0)=0, Df(0,0)=0$ and the Hessian of $f$ at $(0,0)$ is of rank $2$. Then there exists coordinates in an appropriately small neighborhood of $(0,0)$ such that $$f(u,v)=u^2+v^2.$$

I was trying to apply this proposition to the following example to check its validity:

$f(u,v)=uv+u^2v.$ Here $f(0,0)=0,\ Df(0,0)=0 $ and the Hessian is $\begin{pmatrix}0 & 1 \\ 1 & 0. \end{pmatrix}.$ So it satisfy all the hypothesis. Now how can we make change of coordinate such that we will get $f(u,v)=u^2+v^2, $ around some neighborhood of $(0,0)$?

Answer 1

Maybe this factorization helps:

• You have: $(u+u^2)\cdot v$.
• You want: $(u+iv)\cdot (u-iv)$.

So the new $u$ is half the sum of the original factors, the new $v$ is $\frac1{2i}$ times their difference, i.e., $$f(u,v)= \left(\frac{u+u^2+v}{2}\right)^2 +\left(\frac{u+u^2-v}{2i}\right)^2 $$