Analytic Stratification in the Pfaffian Closure of an O-minimal Structure

Introduction. Let U ⊆ R be open and ω = a1dx1+·· ·+andxn a nonsingular, integrable 1-form on U of class C1, and let be the foliation on U associated to ω. A leaf L⊆ U of is a Rolle leaf if any C1 curve γ : [0,1] → U with γ (0),γ (1) ∈ L is tangent to at some point, that is, ω(γ (t))(γ ′(t)) = 0 for some t ∈ [0,1]. Note that while a leaf of is in general only an immersed manifold, any Rolle leaf of is an embedded and closed submanifold of U . Throughout this paper, we fix an arbitrary o-minimal expansion R̃ of the field of real numbers. Whenever U and a1, . . . ,an are definable in R̃ , then a leaf of is called a leaf over R̃ . We use R̃1 to denote the expansion of R̃ by all Rolle leaves over R̃ . For example, the expansion Ran of the real field generated by all globally semianalytic sets is o-minimal; in fact the sets definable in Ran are exactly the globally subanalytic sets (see [7], [4]). Building on Khovanskiı̆’s theory of fewnomials [10] and subsequent work by Moussu and Roche [14], Lion and Rolin [12] showed that (Ran)1 is also o-minimal. Adapting the various ideas involved to the general o-minimal setting, Speissegger [15] proved the following statement.