Keywords
model theory
differential fields
difference fields
Lógica matemática
teoría de modelos
campos diferenciales
campos de diferencia
How to Cite
Abstract
Hrushovski showed that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. We give an axiomatization for DCFA and prove some important model-theoretic results as supersimplicity and elimination of imaginaries. We mention some properties of the fixed field and the constant field of a model of DCFA.
References
Chatzidakis, Z.; Hrushovski, E. (1999) “Model theory of difference fields”, Transactions of the American Mathematical Society 351(8): 2997–3071.
Chatzidakis, Z.; Pillay, A. (1998) “Generic structures and simple theories”, Annals of Pure and Applied Logic 95(1-3): 71–92.
Cohn, R.M. (1965) Difference Algebra. Interscience Publishers John Wiley & Sons, New York-London-Sydney.
Cohn, R.M. (1969) “Systems of ideals”, Canad. J. Math. 21: 783–807.
Cohn, R.M. (1970) “A difference-differential basis theorem”, Canad. J. Math. 22: 1224–1237.
Kim, B.; Pillay A. (1997) ”Simple theories”, Annals of Pure Applied Logic, 88(2-3):149-164.
Kolchin, E.R. (1973) Differential Algebra and Algebraic Groups. Pure and Applied Mathematics, Vol. 54, Academic Press, New York.
Lang, S. (1965) Algebra. Addison-Wesley Publishing Co., Inc., Reading, Massachusets.
Lejeune, H. (1995) Paires de Corps P.A.C. Parfaits, Paires de Corps Pseudofinis. Thèse de Doctorat, Université de Paris 7, Paris.
Marker, D.; Messmer, M.; Pillay, A. (1996) Model Theory of Fields, volume 5 of Lecture Notes in Logic. Springer-Verlag, Berlin.
Marker, D. (2000) “Model theory of differential fields”, in: Haskell, D. Pillay, A. Steinhorn, C. (Eds.) Model Theory, Algebra, and Geometry, volume 39 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge: pages 53–63.
Pierce, D.; Pillay, A. (1998) “A note on the axioms for differentially closed fields of characteristic zero”, J. Algebra 204(1): 108–115.
Pillay, A.; Polkowska, D. (2004) “On PAC and bounded substructures of a stable structure”, Preprint, University of Illinois at Urbana-Champaign, Illinois.
Poizat, B. (1985) Cours de Théorie des Modèles. Une introduction à la logique mathématique contemporaine. Bruno Poizat, Lyon.
van den Dries, L.; Schmidt, K. (1984) “Bounds in the theory of polynomial rings over fields. A nonstandard approach”, Invent. Math. 76(1): 77–91.
Wood, C. (1998) “Differentially closed fields”, in: E. Bouscaren (Ed.) Model Theory and Algebraic Geometry, volume 1696 of Lecture Notes in Math., Springer, Berlin: pages 129–141.