TITLE: Elementary proof of growth estimate for a polynomial via size from its zero set
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.