Toward a Model Theory for Transseries

The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field, and report on our efforts to understand its first-order theory.

[1]  G. A. Edgar.  Transseries for beginners , 2008, 0801.4877.

[2]  Joris van der Hoeven,et al.  Transseries and Real Differential Algebra , 2006 .

[3]  Matthias Aschenbrenner,et al.  H-fields and their Liouville extensions , 2002 .

[4]  A. Wilkie.  Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function , 1996 .

[5]  Angus Macintyre,et al.  Logarithmic‐Exponential Power Series , 1997 .

[6]  S. Kochen,et al.  Diophantine Problems Over Local Fields: III. Decidable Fields , 1966 .

[7]  J. Ressayre,et al.  La Théorie des Modèles, et un Petit Problème de Hardy , 2000 .

[8]  Bernd I. Dahn,et al.  Notes on exponential-logarithmic terms , 1987 .

[9]  Randolph B. Tarrier,et al.  Groups , 1973 .

[10]  P. Bois-Reymond.  Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen , 1875 .

[11]  Lou van den Dries,et al.  T-Convexity and Tame Extensions , 1995, J. Symb. Log..

[12]  M. Potter.  Orders of infinity , 2004 .

[13]  Angus Macintyre,et al.  Logarithmic-exponential series , 2001, Ann. Pure Appl. Log..

[14]  Joris van der Hoeven,et al.  Differentially algebraic gaps , 2005 .

[15]  S. Shelah.  Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory , 1971 .

[16]  Émile Borel,et al.  Mémoire sur les séries divergentes , 1899 .

[17]  L. Dries,et al.  Liouville closed H-fields , 2005 .

[18]  S. Kochen,et al.  Diophantine Problems Over Local Fields I , 1965 .

[19]  Thomas Scanlon,et al.  A model complete theory of valued D-fields , 2000, Journal of Symbolic Logic.

[20]  M. Boshernitzan,et al.  New “orderes of infinity” , 1982 .

[21]  Antongiulio Fornasiero.  Tame structures and open cores , 2010 .

[22]  Thomas Scanlon,et al.  Model theory of the Frobenius on the Witt vectors , 2007 .

[23]  Stability , 1973 .

[24]  Lou van den Dries,et al.  Closed asymptotic couples , 2000 .

[25]  J. Pier.  Development of mathematics , 1994 .

[26]  Ya'acov Peterzil,et al.  Groups, measures, and the NIP , 2006 .

[27]  M. Rosenlicht.  Asymptotic Solutions of Y"=F(x)Y , 1995 .

[28]  M. Rosenlicht,et al.  The rank of a Hardy field , 1983 .

[29]  Jean Ecalle,et al.  Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac , 1992 .

[30]  Saharon Shelah,et al.  Exponentiation in power series fields , 1996 .

[31]  Maxwell Rosenlicht.  On the Value Group of a Differential Valuation II , 1979 .

[32]  Lou van den Dries,et al.  Asymptotic differential algebra , 2005 .

[33]  Nicolas Guzy,et al.  Topological differential fields , 2010, Ann. Pure Appl. Log..

[34]  L. Lipshitz,et al.  Power series solutions of algebraic differential equations , 1984 .

[35]  Lou van den Dries,et al.  Elementary theory of valued fields with a valuation-preserving automorphism , 2010, Journal of the Institute of Mathematics of Jussieu.

[36]  Joris van der Hoeven,et al.  Operators on generalized power series , 2001 .