Global Hölderian Error Bound for Nondegenerate Polynomials

Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be a polynomial and $S = \{x \in \mathbb{R}^n: f(x) \le 0\}$. Let $f_+(x) = \max\{0,f(x)\}$. If there exist $c > 0, \alpha >0, \beta > 0$ such that $d(x,S) \le c([f(x)]_+^\alpha + [f(x)]_+^\beta)$ for all $x \in \mathbb{R}^n$, where $d(x,S)$ denotes the distance between $x$ and $S$, then we say that $f$ has a global Holderian error bound. We prove that if $f$ is convenient and nondegenerate with respect to its Newton boundary at infinity in the sense of Kouchnirenko [ Invent. Math., 32 (1976), pp. 1--32], then it has a global Holderian error bound. Since the nondegenerate polynomials form a Zariski open subset in a variety of polynomials with a given convenient Newton boundary at infinity, this result shows that global Holderian error bounds hold for a large class of polynomials, which can be relatively easy recognized.