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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-3,
author = {Sergio Fratarcangeli},
title = {A first-order version of Pfaffian closure},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {229-254},
zbl = {1147.03021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-3}
}
Sergio Fratarcangeli. A first-order version of Pfaffian closure. Fundamenta Mathematicae, Tome 201 (2008) pp. 229-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-3/