Compactness criteria for real algebraic sets and Newton polyhedra

Abstract

Abstract Let f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}} be a polynomial and 𝒵 ⁢ ( f ) {\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 𝒵 ⁢ ( f ) {\mathcal{Z}(f)} . From this we derive necessary and sufficient criteria for the stable compactness of 𝒵 ⁢ ( f ) {\mathcal{Z}(f)} .