Abstract
Recent developments in the theory of pfaffian sets are presented from a model-theoretic point of view. In particular, the current state of affairs for Van den Dries’s model-completeness conjecture is discussed in some detail. I prove the o-minimality of the pfaffian closure of an o-minimal structure, and I extend a weak model completeness result, originally proved as Theorem 5.1 in (J.-M. Lion and P. Speissegger, Duke Math J 103:215–231, 2000), to certain reducts of the pfaffian closure, such as the reduct generated by a single pfaffian chain.
Mathematics Subject Classification (2010): Primary 14P15, 58A17, Secondary 03C64
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References
J. Bochnak, M. Coste, M.-F. Roy, Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)), vol. 36 (Springer, Berlin, 1998). Translated from the 1987 French original, Revised by the authors
C. Camacho, A. Lins Neto, Geometric Theory of Foliations (Birkhäuser Boston Inc., Boston, 1985). Translated from the Portuguese by Sue E. Goodman
A. Gabrielov, Complements of subanalytic sets and existential formulas for analytic functions. Invent. Math. 125, 1–12 (1996)
A. Gabrielov, Relative Closure and the Complexity of Pfaffian Elimination, in Discrete and computational geometry. Algorithms Combin, vol. 25 (Springer, Berlin, 2003), 441–460
A. Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes. Comment. Math. Helv. 32, 248–329 (1958)
G.O. Jones, Zero sets of smooth functions in the Pfaffian closure of an o-minimal structure. Proc. Am. Math. Soc. 136, 4019–4025 (2008)
M. Karpinski, A. Macintyre, Polynomial bounds for VC dimension of sigmoidal and general Pfaffian neural networks. J. Comput. Syst. Sci. 54, 169–176 (1997). 1st Annual Dagstuhl Seminar on Neural Computing (1994)
A.G. Khovanskiĭ, Real analytic manifolds with the property of finiteness, and complex abelian integrals. Funktsional. Anal. i Prilozhen. 18, 40–50 (1984)
A.G. Khovanskiĭ, Fewnomials. Translations of Mathematical Monographs, vol. 88 (American Mathematical Society, Providence, 1991). Translated from the Russian by Smilka Zdravkovska
K. Kuratowski, Topology, vol. I, new edition, revised and augmented (Academic, New York, 1966). Translated from the French by J. Jaworowski
K. Kuratowski, Topology, vol. II, new edition, revised and augmented. (Academic, New York, 1968). Translated from the French by A. Kirkor
J.-M. Lion, Inégalité de Lojasiewicz en géométrie pfaffienne. Ill. J. Math. 44, 889–900 (2000)
J.-M. Lion, Algebraic measure, foliations and o-minimal structures, in o-minimal Structures, Proceedings of the RAAG Summer School Lisbon 2003, ed. by M.J. Edmundo, D. Richardson, J. Wilkie (Cuvillier Verlag, Göttingen, Germany, 2005), pp. 159–171
J.-M. Lion, J.-P. Rolin, Volumes, feuilles de Rolle de feuilletages analytiques et théorème de Wilkie. Ann. Fac. Sci. Toulouse Math. 7(6), 93–112 (1998)
J.-M. Lion, P. Speissegger, Analytic stratification in the Pfaffian closure of an o-minimal structure. Duke Math. J. 103, 215–231 (2000)
J.-M. Lion, P. Speissegger, A geometric proof of the definability of Hausdorff limits. Selecta Math. (N.S.) 10, 377–390 (2004)
J.-M. Lion, P. Speissegger, Un théorème de type Haefliger définissable. Astérisque 323, 197–221 (2009)
J.-M. Lion, P. Speissegger, The theorem of the complement for nested sub-Pfaffian sets. Duke Math. J. 155, 35–90 (2010)
J.-M. Lion, C. Miller, P. Speissegger, Differential equations over polynomially bounded o-minimal structures. Proc. Am. Math. Soc. 131, 175–183 (2003)
C. Miller, Basics of o-minimality and Hardy Fields. Fields Institute Communications, Springer Verlag, 62, 43–69 (2012)
R. Moussu, C. Roche, Théorie de Hovanskiĭ et problème de Dulac. Invent. Math. 105, 431–441 (1991)
P. Speissegger, The Pfaffian closure of an o-minimal structure. J. Reine Angew. Math. 508, 189–211 (1999)
M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. I, 2nd edn. (Publish or Perish Inc, Wilmington, 1979)
L. van den Dries, 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. Am. Math. Soc. 9, 1051–1094 (1996)
A.J. Wilkie, A theorem of the complement and some new o-minimal structures. Sel. Math. (N.S.) 5, 397–421 (1999)
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Supported by the Fields Institute for Research in the Mathematical Sciences and by NSERC of Canada grant RGPIN 261961.
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Speissegger, P. (2012). Pfaffian Sets and O-minimality. In: Miller, C., Rolin, JP., Speissegger, P. (eds) Lecture Notes on O-Minimal Structures and Real Analytic Geometry. Fields Institute Communications, vol 62. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4042-0_5
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