Suppose you are given a convex polyhedron $\Delta$ in $\mathbb{R}^n$ (i.e. a convex hull of finitely many points in $\mathbb{Z}^n$) and consider a finite dimensional vector space $V$ over $\mathbb{C}$ defined by $$V=\left\{f(x)=\sum_{\alpha\in\mathbb{Z}^n}a_\alpha x^\alpha\in\mathbb{C}[x_1^{\pm},\cdots,x_n^\pm] \mid a_\alpha=0 \text{ if } \alpha\notin\Delta \right\}.$$ Suppose further that $\dim \Delta\leq n-1.$ We say $f\in V$ is Newton non-degenerate if $$\left\{x\in(\mathbb{C}^{*})^n\mid\frac{\partial f}{\partial x_1}(x)=\cdots=\frac{\partial f}{\partial x_n}(x)=0\right\}=\emptyset.$$ My question is

Is the set $$\{f\in V|f\text{ is Newton non-degenerate}\}$$ Zariski open in $V$?.

It is widely believed that this result was first affirmatively proved by Kouchnirenko (Theorem 6.1., Invent. Math., 1976; link), but it seems he just proved there is a Zariski open subset of $V$ consisting only of Newton non-degenerate polynomials. It might be possible that my understanding of French is not good enough, but at least he does not prove the Zariski openness of the set $\{f\in V\mid f\text{ is Newton non-degenerate}\}$.

Could you please give me another reference stating that the set $\{f\in V|f\text{ is Newton non-degenerate}\}$ is Zariski open? Or could you suggest the outline of the proof?

Thank you very much in advance