The Pfaffian Closure: Analytic Stratification

. Let e R be an o-minimal expansion of the field of real numbers. We show that if e R has analytic cell decomposition, then its Pfaffian closure P \Gamma e R \Delta also has analytic cell decomposition. In particular, if e R has analytic Whitney stratification, then so does P \Gamma e R \Delta . 1. Introduction Let U ` R n be open and ! = a 1 dx 1 + \Delta \Delta \Delta + a n dx n a nonsingular 1-form on U of class C 1 . A leaf L ` U of ! is a Rolle leaf if L is an embedded and closed submanifold of U , and if any C 1 curve fl : (0; 1) \Gamma! U that intersects L in at least two points is tangent to ! at some point, that is, !(fl(t))(fl 0 (t)) = 0 for some t 2 (0; 1). Throughout this paper we fix an arbitrary o-minimal expansion e R of the field of real numbers. If U and a 1 ; : : : ; a n are definable in e R , then any leaf of ! is called a leaf over e R . For example, the expansion R an of the real field generated by all globally semianalytic sets is o-mini..