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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

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  • $\begingroup$ To me this question feels garbled (my fault?). $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 4, 2013 at 0:28

1 Answer 1

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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.

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  • $\begingroup$ Thank you. But i think that some thing is missing.Let the degree of polynomial is n. Then we need to know, on the unit spher, \delta is in the for O(\epslon^{n}) $\endgroup$
    – Ali Taghavi
    Commented Sep 5, 2013 at 16:43
  • $\begingroup$ I think that the order power in Lojasiewicz type inequalities is sensitive in this question. So do you think that the Ulam stability of homogeneous polynomials in at least 3 variables can be solved with other approaches? however in 2 variable the subject is obvious because of factorization to polynomials with low(1 or 2) degree $\endgroup$
    – Ali Taghavi
    Commented Sep 5, 2013 at 18:10

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