Let $P$ be a homogenous polynomial with real coefficients in several variable(at least three variable)
Is the following statement true:



> 
> For every $\epsilon$ there is a $\delta$ such that for every x with $|P(x)|< \delta$ we have $d(x,Z)<\epsilon$.
> 
> 
> 


Here $Z=P^{-1}(\{0\})$ is the set of roots of $P$, and $d$ is the standard .distance



Ulam stability of homogeneous polynomials
EDIT. My previous answer was incorrect. So I replace it.
The answer is no. A counterexample is
$$y^{2m}+(z^{m-1}y-x^m)^2.$$
This is of degree $2m$ but $\delta$ is like $\epsilon^{2m^2}$ near the
point $(0,0,1)$.


I found this example in the paper of Kollar and Shiffman, TAMS 329 (1992), on the very first page. They credit it to Lojasiewicz himself, IHES, Bures-sur-Yvette, 1965.
I do not know how to find this IHES preprint, but Kollar and Shiffman is available
to everyone online.


