Keywords
Supersimple Theories
Difference-Differential Fields
Definable Sets
Teoría de modelos
teorías supersimples
campos diferenciales de diferencia
conjuntos definibles
How to Cite
Abstract
This is the first of two papers devoted to the proof of Zilber’s dichotomy for the case of difference-differential fields of characteristic zero. In this paper we use the techniques exposed in [9] to prove a weaker version of the dichotomy, more precisely, we prove the following: in DCFA the canonical base of a finite-dimensional type is internal to the fixed field of the field of constants. This will imply a weak version of Zilber’s dichotomy: a finite-dimensional type of SU -rank 1 is either 1-based or non-orthogonal to the fixed field of the field of constants.
References
Bustamante Medina, R.F. (2007) “Differentially closed fields of characteristic zero with a generic automorphism”, Revista de Matemática: Teoría y Aplicaciones 14(1): 81–100.
Bustamante Medina, R.F. (2008) “Rank and dimension in difference-differential fields”, Submitted.
Chatzidakis, Z.; Hrushovski, E. (1999) “Model theory of difference fields”, Transactions of the American Mathematical Society 351(8): 2997–3071.
Eisenbud, D. (1995) Commutative Algebra. Springer-Verlag, New York.
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: D. Haskell, A. Pillay & C. Steinhorn (Eds.) Model Theory, Algebra, and Geometry, volume 39 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge: 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. (2002) “Model-theoretic consequences of a theorem of Campana and Fujiki”, Fundamenta Mathematicae 174(2): 187–192.
Pillay, A.; Ziegler, M. (2003) “Jet spaces of varieties over differential and difference fields”, Selecta Mathematica. New Series 9(4): 579–599.
Shafarevich, I.R. (1977) Basic Algebraic Geometry. Springer-Verlag, Berlin