Fedoryuk values and stability of global H\"{o}lderian error bounds for polynomial functions

Abstract

Let $f$ be a polynomial function of $n$ variables. In this paper, we study stability of global H\"{o}lderian error bound for a nonempty sublevel set $[f \le t]$ under a perturbation of $t$. In this paper, we give: * Criteria for the existence of a global H\"{o}lderian error bound of $[f \le t]$; * Formulas for computing explicitly the set $$H(f) := \{ t \in \mathbb{R}: [f \le t]\ \text{has a global H\"{o}lderian error bound}\}$$ via some Fedoryuk values of $f$ and definition of threshold for the existence of global H\"{o}lderian error bound of $f$; * Definition of all types of stability of global H\"{o}lderian error bound of $[f \le t]$.