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?