On gradient at infinity of semialgebraic functions

Abstract

Let $f: \mathbb{R}^n \to\mathbb{R}$ be a $C^2$ semialgebraic function and let $c$ be an asymptotic critical value of $f$. We prove that there exists a smallest rational number $\varrho_c\leq 1$ such that $|x|\cdot |\nabla f|$ and $|f(x)-c|^{\varrho_c}