Let $\Omega = (\omega_{1}, \dots, \omega_{n-k})$ be differential $1$-forms with polynomial coefficients in ${\bf R}^{n}$. A
Pfaffian manifold
of $\Omega$ is by definition
a maximal integral $k$-manifold of $\Omega$. It is shown that the number of homeomorphism classes of all Pfaffian manifolds of Rolle Type of $\Omega$ is finite and, moreover, bounded by a computable function in variables $n$, $k$ and the degree of $\omega_{1}, \dots, \omega_{n-k}$. Finiteness is proved also in any o-minimal structure.
We give also an example of a semi-algebraic $C^{1}$ differential form on a semialgebraic $C^{2}$ $3$-manifold whose Pfaffian manifolds have homeomorphism classes of the cardinality of continuum. Hence the cardinality of all manifolds is the continuum (not countable).
References
S. Basu, R. Pollack and M.-F. Roy, On the combinatorial and algebraic complexity of quantifier elimination , Journal of ACM, 43 , no. 6, 1002--1045 (1996).
F. Chazal, Sur les feuilletages algébriques de Rolle , Comment. Math. Helv., 72 , 411--425 (1997).
F. Chazal, Strcture locale et globale des feuilletage de Rolle, un théorème de fibration , Ann. Inst. Fourier, 48 , no. 2, 553--592 (1998).
G. E. Collins, Quantifier elimination for real closed fields by cylinderical algebraic decomposition , Springer Lecture Notes in Computer Science 33, 515--532.
Mathematical Reviews (MathSciNet):
MR403962
A. Haefliger, Structures feuilletées et cohomologie á valeurs dans un faisceau de groupöide , Comment. Math. Helv., 32 , 248--329 (1958).
Mathematical Reviews (MathSciNet):
MR100269
A. Khovanskii, On a class of systems of trancendental equations , Soviet Math. Dokl., 22 , no. 3, 762--765 (1980).
J.-M. Lion, Étude des hypersurfaces Pfaffiennes de Rolle , PhD thesis, Université de Bourgogne (1991).
J.-M. Lion and J.-P. Rolin, Volumes, feuilles de Rolle et feuilletages analytiques réeles et théorème de Wilkie , Ann. Toulouse, 7 , 93--112 (1998).
J.-M. Lion and P. Speissegger, Analytic strafication in the pfaffian closure of an o-minimal structure , Duke Math. J., 103 , no. 2, 215--231 (2000).
R. Moussu and C. Roche, Théorie de Khovanskii et problème de Dulac , Invent. Math., 105 , 431--441 (1991).
R. Moussu and C. Roche, Théorème de finitude uniformes pour les variétés pfaffiennes de Rolle , Ann. Inst. Fourier, 42 , 393--420 (1992).
M. Shiota, Nash manifolds, Lecture Notes in Mathematics 1269, Springer-Verlag (1980).
Mathematical Reviews (MathSciNet):
MR904479
M. Shiota, Geometry of subanalytic and semialgebraic sets, Birkhäuser (1997).
P. Speissegger, The Pfaffian closure of an o-minimal structure , J. reine angew. Math., 508 , 189--221 (1999).
L. van den Dries, Tame topology and O-minimal structure, London Math. Soc. Lecture Note 248, Cambridge Univ. Press (1998).
L. van den Dries and C. Miller, Geometric categories and o-minimal structures , Duke Math. J., 84 , 497--540 (1996).
A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function , J. Amer. Math. Soc., 9 , 1051--1094 (1996).
