The theorem of the complement for nested sub-Pfaffian sets

Jean-Marie Lion; Patrick Speissegger

doi:10.1215/00127094-2010-050

Duke Mathematical Journal

Duke Math. J.
Volume 155, Number 1 (2010), 35-90.

The theorem of the complement for nested sub-Pfaffian sets

Jean-Marie Lion and Patrick Speissegger

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Abstract

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 $\curly{L}_{\rm nest}(\curly{R})$ .

Article information

Source
Duke Math. J., Volume 155, Number 1 (2010), 35-90.

Dates
First available in Project Euclid: 23 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1285247218

Digital Object Identifier
doi:10.1215/00127094-2010-050

Mathematical Reviews number (MathSciNet)
MR2730372

Zentralblatt MATH identifier
1226.14075

Subjects
Primary: 14P10: Semialgebraic sets and related spaces 58A17: Pfaffian systems
Secondary: 03C99: None of the above, but in this section

Citation

Lion, Jean-Marie; Speissegger, Patrick. The theorem of the complement for nested sub-Pfaffian sets. Duke Math. J. 155 (2010), no. 1, 35--90. doi:10.1215/00127094-2010-050. https://projecteuclid.org/euclid.dmj/1285247218

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