Improved effective Łojasiewicz inequality and applications

Abstract

Let $\mathrm {R}$ be a real closed field. Given a closed and bounded semialgebraic set $A \subset \mathrm {R}^n$ and semialgebraic continuous functions $f,g:A \rightarrow \mathrm {R}$ such that $f^{-1}(0) \subset g^{-1}(0)$ , there exist an integer $N> 0$ 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 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 $\mathcal {Q}$ . The previous best-known upper bound in this generality appeared in P. Solernó, Effective Łojasiewicz Inequalities in Semi-Algebraic Geometry, Applicable Algebra in Engineering, Communication and Computing (1991) and depended on the sum of degrees of the polynomials defining $A,f,g$ and thus implicitly on 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, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common upper bound on the Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions. This improves a prior result of Chris Miller (C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic (1994)).

Keywords

• 14P10
• 03C64
• 90C23