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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 $f$.

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By "free term", do you mean constant term? Also, I think it would be easier to read the stuff after the word "where" if you used bullet points. [To go to the next line, type two spaces before an enter/return.] –  Charles Staats May 31 '10 at 22:59
    
The tag ag.algebraic-geometry is inappropriate, this is analysis. –  Roland Bacher Jun 1 '10 at 6:58
    
@Roland I think, this area is called real algebraic geometry. Wiki thinks the same way en.wikipedia.org/wiki/%C5%81ojasiewicz_inequality. –  Fedor Petrov Jun 1 '10 at 8:42

1 Answer 1

up vote 2 down vote accepted

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.

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