A first-order version of Pfaffian closure

Sergio Fratarcangeli¹

¹ Division of Natural Sciences and Mathematics The College of New Rochelle 29 Castle Place New Rochelle, NY 10805, U.S.A.

Fundamenta Mathematicae, Tome 198 (2008) no. 3, pp. 229-254

Résumé

The purpose of this paper is to extend a theorem of Speissegger [J. Reine Angew. Math. 508 (1999)], which states that the Pfaffian closure of an o-minimal expansion of the real field is o-minimal. Specifically, we display a collection of properties possessed by the real numbers that suffices for a version of the proof of this theorem to go through. The degree of flexibility revealed in this study permits the use of certain model-theoretic arguments for the first time, e.g. the compactness theorem. We illustrate this advantage by deriving a uniformity result on the number of connected components for sets defined with Rolle leaves, the building blocks of Pfaffian-closed structures.

Détail

DOI : 10.4064/fm198-3-3

Keywords: purpose paper extend theorem speissegger reine angew math which states pfaffian closure o minimal expansion real field o minimal specifically display collection properties possessed real numbers suffices version proof theorem through degree flexibility revealed study permits certain model theoretic arguments first time compactness theorem illustrate advantage deriving uniformity result number connected components sets defined rolle leaves building blocks pfaffian closed structures