The theorem of the complement for sub-Pfaffian sets

Jean-Marie Lion (1) , Patrick Speissegger (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Department of Mathematics & Statistics [Halmiton]

Let R be an o-minimal expansion of the real field, and let P(R) be its Pfaffian closure. Let L be the language consisting of all Rolle leaves added to R to obtain P(R). We prove that P(R) is model complete in the language L, provided that R admits analytic cell decomposition. We do this by proving a somewhat stronger statement, the theorem of the complement for nested sub-Pfaffian sets over R. As a corollary, we obtain that P(R) is obtained by adding to R all nested Rolle leaves over R, a one-stage process.

Mots clés: O-minimal structures; Pfaffian systems; analytic stratification

Domaines: Géométrie différentielle [math.DG]; Logique [math.LO]

Jean-Marie Lion, Patrick Speissegger. The theorem of the complement for sub-Pfaffian sets. Duke Mathematical Journal, 2010, 155 (1), pp.35-90. ⟨hal-00008450⟩