In this section, we recall some results concerning stably steep polynomials, which were defined in Definition \[stabsteep\]. The first one states that stably steep polynomials of sufficiently high degree are generic, and was used in the justification of Corollary \[coro1\] and Corollary \[coro2\].

\[propNekho2\] The complement of $SS(n,m_0)$ in $P_2(n,m_0)$ is contained in a closed semi-algebraic subset of codimension at least one. In particular, $SS(n,m_0)$ is an open dense set of full Lebesgue measure.

For a proof, we refer to Proposition A.2 and Proposition A.3 in [@BFN15]. In order to state the next results, we first recall the definition of steep functions.

\[funcsteep\] A differentiable function $h : G \rightarrow {\mathbb{R}}$ is steep on a domain $G' \subseteq G$ if there exist positive constants $C,\delta,p_l$, for any integer $l\in [1,n-1]$, and $\kappa$ such that for all $I \in G'$, we have $||\nabla h(I)|| \geq \kappa$ and, for all integer $l\in [1,n-1]$, for all vector space $\Lambda \in {\mathbb{R}}^n$ of dimension $l$, letting $\lambda=I+\Lambda$ the associated affine subspace passing through $I$ and $h_\lambda$ the restriction of $h$ to $\lambda$, the inequality $$\max_{0 \leq \eta \leq \xi}\;\min_{||I'-I||=\eta, \; I' \in \lambda}||\nabla h_\lambda(I')-\nabla h_\lambda(I)||>C\xi^{p_l}$$ holds true for all $0 < \xi \leq\delta$. We 