Duke Mathematical Journal

The theorem of the complement for nested sub-Pfaffian sets

Jean-Marie Lion and Patrick Speissegger

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Abstract

Let R be an o-minimal expansion of the real field, and let Lnest(R) be the language consisting of all nested Rolle leaves over R. We call a set nested sub-Pfaffian over R if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over R is again a nested sub-Pfaffian set over R. As a corollary, we obtain that if R admits analytic cell decomposition, then the Pfaffian closure P(R) of R is obtained by adding to R all nested Rolle leaves over R, a one-stage process, and that P(R) is model complete in the language Lnest(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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