Differential Algebra and Related Topics: Proceedings of the by P. Cassidy

By P. Cassidy

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Extra info for Differential Algebra and Related Topics: Proceedings of the International Workshop, Newark Campus of Rutgers, The State University of New Jersey, 2-3 November 2000

Example text

It is easy to see t h a t if A has the strong Rosenfeld property, then it has the Rosenfeld property since the initials I A are partially reduced with respect to A . If A has the Rosenfeld property, then it is possible to answer certain questions about a differential ideal by answering similar questions about an ideal. For the differential ideals a # and as, we shall be interested in t h e properties of being prime, radical, and zero-reduced. 3, an autoreduced subset of a proper differential ideal a is a characteristic set if o is zero-reduced with respect to it.

His lemma then states that a coherent autoreduced set has the Rosenfeld property. Kolchin [23] (Chapter III, Section 8, p. 135) generalized this to L-coherence relative to an ideal L (not necessarily differential, but having a set of generators partially reduced with respect to A) and over differential domains of arbitrary characteristic. Morrison [29] recently introduced new 10 This notion is closely related to, and probably implies, the involutiveness, formal integrability, or other completeness properties of the system defined by A .

1 For any v 6 @Y, let A(„) be the set of all differential polynomials 6A with 6 € ©, A G A and 9UA < v. We say an autoreduced set A is coherent (resp. subcoherent11, resp. strongly coherent, resp. strongly subcoherent) if for all pairs A, A' £ A whose leaders UA,UA> have a common derivative, say v = QUA = O'UA', the differential polynomial A(A,A',v) = SA>0A — SAO'A', which has lower rank than v, belongs to the ideal (A(„)): H°° (resp. (A U A ( „)): H°°, resp. (A ( „ } ): 5°°, resp. (A U A ( „)): 5°°) in the polynomial ring 01.

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