Lojasiewicz inequality in Pfaffian geometry (Q1593037)

The Rolle leaves, which are leaves of special real analytic foliations of codimension one, have an interesting finiteness property: Pfaffian sets which are an intersection of a finite number of relatively compact Rolle leaves have a finite number of connected components [see the paper of \textit{R. Moussu} and \textit{C. A. Roche}, Invent. Math. 105, 431-441 (1991; Zbl 0769.58050)]. \textit{J. M. Lion} and \textit{J. P. Rolin} [Ann. Fac. Sci. Toulouse 7, 93-112 (1998; Zbl 0933.32014)] proved that the Rolle leaves which are relatively compact are the elements of an o-minimal structure: the family \(T^\infty\) of \(T^\infty\)-Pfaffian sets. Hence these subsets verify a Łojasiewicz inequality. The main goal of this work is to make this inequality more precise: Let \(A\) and \(B\) two \(T^\infty\)-Pfaffian compact sets of \({\mathbb R}^n\). Then there is a positive integer \(N\) such that \[ 1/\exp_N (1/d(x,A \cap B)) \leq d(x,B) \] whenever \(x \in A \setminus B\). The \(N\)th iteration of the exponential is denoted by \(\exp_N\). It must be noticed that this statement is optimal since the closure of \(\{(x,1/\exp_N (1/x))\mid x \in ]0,1] \}\) is a \(T^\infty\)-Pfaffian subset.