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Superstable differential fields

Published online by Cambridge University Press:  12 March 2014

A. Pillay  and
Ž. Sokolović
A. Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, E-mail: [email protected]
Ž. Sokolović
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, E-mail: [email protected]
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Extract

In this paper we study differential fields of characteristic 0 (with perhaps additional structure) whose theory is superstable. Our main result is that such a differential field has no proper strongly normal extensions in the sense of Kolchin [K1]. This is an approximation to the conjecture that a superstable differential field is differentially closed (although we believe the full conjecture to be false). Our result improves earlier work of Michaux [Mi] who proved that a (plain) differential field with quantifier elimination has no proper Picard-Vessiot extension. Our result is a generalisation of Michaux's, due to the fact that any plain differential field K with quantifier elimination is ω-stable. (Any quantifier free type over K defines a unique type over K in the sense of dc(k), the differential closure of K, and as we mention below the theory of differentially closed fields is ω-stable.) The proof of our main result depends on (i) Kolchin's theory [K3] which states that any strongly normal extension L of an algebraically closed differential field K is generated over K by an element η of some algebraic group G defined over CK, the constants of K, where η satisfies some specific differential equations over K related to invariant differential forms on G (η is “G-primitive” over K), and (ii) the fact that a superstable field has a unique generic type which is semiregular.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 1 , March 1992 , pp. 97 - 108
Copyright
Copyright © Association for Symbolic Logic 1992

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References

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