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  • Online publication date: March 2017

Remarks on algebraic D-varieties and the model theory of differential fields


Abstract.We continue our investigation of the category of algebraic D-varieties and algebraic D-groups from [5] and [9]. We discuss issues of quantifier-elimination, completeness, as well as the D-variety analogue of “Moishezon spaces”.

Introduction and preliminaries.If (K, ∂) is a differentially closed field of characteristic 0, an algebraic D-variety over K is an algebraic variety over K together with an extension of the derivation to a derivation of the structure sheaf of X. Welet D denote the category of algebraic D-varieties and G the full subcategory whose objects are algebraic D-groups. D and G were essentially introduced by Buium and G was exhaustively studied in [2]. The category D is closely related but not identical to the class of sets of finite Morley rank definable in the structure (K,+, ・, ∂) (which is essentially the class of “finitedimensional differential algebraic varieties”). However an object ofD can also be considered as a first order structure in its own right by adjoining predicates for algebraic D-subvarieties of Cartesian powers. In a similar fashion D can be considered as a many-sorted first order structure. In [5] it was shown that the many-sorted “reduct” G has quantifier-elimination. We point out in this paper an easy example showing:

PROPOSITION 1.1. D does not have quantifier-elimination.

The notion of a “complete variety” in algebraic geometry is fundamental. Over C these are precisely the varieties which are compact as complex spaces. Kolchin [4] introduced completeness in the context of differential algebraic geometry. This was continued by Pong [12] who obtained some interesting results and examples. We will give a natural definition of completeness for algebraic D-varieties. The exact relationship to the notion studied by Kolchin and Pong is unclear: in particular I do not know whether any of the examples of new “complete” -closed sets in [12] correspond to complete algebraic D varieties in our sense.

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