The theorem of the complement for nested sub-Pfaffian sets
Lion, Jean-Marie ; Speissegger, Patrick
Duke Math. J., Tome 151 (2010) no. 1, p. 35-90 / Harvested from Project Euclid
Let $\curly{R}$ be an o-minimal expansion of the real field, and let $\curly{L}_{\rm nest}(\curly{R})$ be the language consisting of all nested Rolle leaves over $\curly{R}$ . We call a set nested sub-Pfaffian over $\curly{R}$ if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over $\curly{R}$ . Assuming that $\curly{R}$ admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over $\curly{R}$ is again a nested sub-Pfaffian set over $\curly{R}$ . As a corollary, we obtain that if $\curly{R}$ admits analytic cell decomposition, then the Pfaffian closure $\curly{P}(\curly{R})$ of $\curly{R}$ is obtained by adding to $\curly{R}$ all nested Rolle leaves over $\curly{R}$ , a one-stage process, and that $\curly{P}(\curly{R})$ is model complete in the language $\curly{L}_{\rm nest}(\curly{R})$ .
Publié le : 2010-10-01
Classification:  14P10,  58A17,  03C99
@article{1285247218,
     author = {Lion, Jean-Marie and Speissegger, Patrick},
     title = {The theorem of the complement for nested sub-Pfaffian sets},
     journal = {Duke Math. J.},
     volume = {151},
     number = {1},
     year = {2010},
     pages = { 35-90},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1285247218}
}
Lion, Jean-Marie; Speissegger, Patrick. The theorem of the complement for nested sub-Pfaffian sets. Duke Math. J., Tome 151 (2010) no. 1, pp.  35-90. http://gdmltest.u-ga.fr/item/1285247218/