Improved effective Łojasiewicz inequality and applications

Abstract

Let $\mathrm{R}$ be a real closed field. Given a closed and bounded semi-algebraic set $A \subset \mathrm{R}^n$ and semi-algebraic continuous functions $f,g:A \rightarrow \mathrm{R}$, such that $f^{-1}(0) \subset g^{-1}(0)$, there exist $N$ and $c \in \mathrm{R}$, such that the inequality (Łojasiewicz inequality) $|g(x)|^N \le c \cdot |f(x)|$ holds for all $x \in A$. In this paper we consider the case when $A$ is defined by a quantifier-free formula with atoms of the form $P = 0, P >0, P \in \mathcal{P}$ for some finite subset of polynomials $\mathcal{P} \subset \mathrm{R}[X_1,\ldots,X_n]_{\leq d}$, and the graphs of $f,g$ are also defined by quantifier-free formulas with atoms of the form $Q = 0, Q >0, Q \in \mathcal{Q}$, for some finite set $\mathcal{Q} \subset \mathrm{R}[X_1,\ldots,X_n,Y]_{\leq d}$. We prove that the Łojasiewicz exponent $N$ in this case is bounded by $(8 d)^{2(n+7)}$. Our bound depends on $d$ and $n$, but is independent of the combinatorial parameters, namely the cardinalities of $\mathcal{P}$ and $\mathcal{Q}$. As a consequence we improve the current best error bounds for polynomial systems under some conditions. Finally, as an abstraction of the notion of independence of the Łojasiewicz exponent from the combinatorial parameters occurring in the descriptions of the given pair of functions, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions.