Is there any easy proof or/and reference for the following 


**Proposition.** If a polynomial $f\in \mathbb{R}[x\_1,\dots,x\_n]$ with zero constant term has isolated local minimum in 0, then $|f|> C (x\_1^2+\dots +x\_n^2)^N$ in some neighborhood of 0, where  

$\bullet$ $N$ depends only on degree of $f$ and $n$,  

$\bullet$ $C$ depends on uniform bound on $n$, degree of $f$, and bounds for coefficients of $f$. 


?



minimum of polynomial
That there is some $C$ and $N$ for which this is satisfied is a case of Lojasiewicz's theorem, which says that if $f(x\_0) = 0$, then there is an open set containing $x\_0$ on which $|f(x)| \geq C|d(x,Z)|^N$ for some $C$ and $N$. Here $d(x,Z)$ denotes the distance from $x$ to the zero set $Z$ of $f(x)$. So in the isolated singularity case it reduces to what you're asking for. 


I don't any immediate way to bound the constants in the way you're asking for.


