Compactness criteria for real algebraic sets and newton polyhedra

Let $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to \mathbb{R}$ be a polynomial and $\mathcal{Z}(f)$ its zero set. In this paper, in terms of the so-called Newton polyhedron of $f,$ we present a necessary criterion and a sufficient condition for the compactness of $\mathcal{Z}(f)\mathrm{.}$ From this we derive necessary and sufficient criteria for the stable compactness of $\mathcal{Z}(f)\mathrm{.}$