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The paper Asymptotic properties of polynomials and algebraic functions of several variables by Gorin contains the following.

Lemma 3.1. Let $f\in \mathbb R[x_1,\dots,x_n]$. Suppose $f$ has a root in the interior of the unit circle. Then there exists a positive constant $c$ such that $$\|f(x)\|\geq c \cdot d(x,f^{-1}(0))^{\deg f},$$ where $d$ denotes the standard metric on $\mathbb R^n$.

The proof involves a previous theorem which is itself rather involved.

Question. Is there a quick direct proof of the above lemma?

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    $\begingroup$ The statement is weird. The stated bound on $\|f(x)\|$ is invariant under shifts, hence the condition on a root in the unit circle should be redundant – you can always shift the polynomial so that e.g. the origin is a root. If $f$ has any root at all, that is. $\endgroup$
    – Emil Jeřábek
    Jun 11, 2019 at 8:22
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    $\begingroup$ @EmilJeřábek I agree the condition is redundant, but it's part of the statement in the reference, so I figured I should write it down. $\endgroup$
    – Arrow
    Jun 11, 2019 at 8:25
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    $\begingroup$ Sure. I just mentioned it because it raises the possibility that there is a typo in the original paper. $\endgroup$
    – Emil Jeřábek
    Jun 11, 2019 at 8:32
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    $\begingroup$ @EmilJeřábek I guess the evaluation point is supposed to be in the disk (or some other compact) too. The statement is clearly false in the entire space. $\endgroup$
    – fedja
    Jun 11, 2019 at 19:57

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The Russian original text clearly states that the exponent is not $\deg f$ but just some $\alpha>0$. For $\alpha=\deg f$ is it not true: a polynomial $f(x,y)=x^{2n}+(x-y^n)^2$ has a unique zero $(0,0)$, but $f(\delta^n,\delta)=\delta^{2n^2}$.

I remember obtaining a cubic upper bound for $\alpha$ in the case of a polynomial in two variables with a single zero (but the estimate holds only in a disk centred at the root, not on the whole plane). I may try to find it if you wish.

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