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?



Elementary proof of growth estimate for a polynomial via size from its zero set
The [Russian original text](http://www.mathnet.ru/links/6d10d251a8c07ce8e60bf7f30ddd3e1d/rm6566.pdf) 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.


