Real Algebraic and Analytic Geometry |

The Theorem of the Complement for nested Sub-Pfaffian Sets.

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Submission: 2009, March 9.

*Abstract:
Let R be an o-minimal expansion of the real field, and let
L(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 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 L(R).*

Mathematics Subject Classification (2000): 14P10, 58A17, 03C99.

Keywords and Phrases: Differential Geometry, Logic, O-minimal structures, Pfaffian systems, analytic stratification.

**Full text**: http://arXiv.org/abs/math/0602196

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