Łojasiewicz exponent near the fibre of a mapping

Abstract:

Let $g:X\to \mathbb {R}^k$ and $f:X\to \mathbb {R}^m$, where $X\subset \mathbb {R}^n$, be continuous semi-algebraic mappings, and $\lambda \in \mathbb {R}^m$. We describe the optimal exponent $\theta =:\mathcal {L}_{\infty ,f\to \lambda }(g)$ for which the Łojasiewicz inequality $|g(x)|\geqslant C|x|^\theta$ holds with $C>0$ as $|x|\to \infty$ and $f(x)\to \lambda$. We prove that there exists a semi-algebraic stratification $\mathbb {R}^m=S_1\cup \cdots \cup S_j$ such that the function $\lambda \mapsto \mathcal {L}_{\infty ,f\to \lambda }(g)$ is constant on each stratum $S_i$. We apply this result to describe the set of generalized critical values of $f$.