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Some remarks concerning the theory of ordered differentially closed fields. (Quelques remarques concernant la théorie des corps ordonnés différentiellement clos.) (French. English summary) Zbl 1104.03028
The authors give a geometrical axiomatization of the closed ordered differential fields [introduced by M. F. Singer, “The model theory of ordered differential fields”, J. Symb. Log. 43, 82–91 (1978; Zbl 0396.03031)], that is analogous to the Pierce-Pillay axiomatization of differentially closed fields of characteristic zero [D. Pierce and A. Pillay, “A note on the axioms for differentially closed fields of characteristic zero”, J. Algebra 204, 108–115 (1998; Zbl 0922.12006)]. They also show that this theory does not have the independence property by using the “forgetful functor” which reduces a quantifier-free ordered-differential formula to the natural ordered-algebraic one, where the derivative \(Dx\) is replaced by a new variable \(y\). The result then follows because real closed fields do not have the independence property. The authors remark that this trick can be used in a number of other theories of differential rings.

03C60 Model-theoretic algebra
12H05 Differential algebra
12J15 Ordered fields
12L12 Model theory of fields