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Differential algebraic function fields depending rationally on arbitrary constants. (English) Zbl 0695.12016

Let K be an ordinary differential field of characteristic zero. A differential field extension L of K is said to depend rationally on arbitrary constants if there exists a differential field extension M of K such that L and M are free over K and \(LM=MC_{LM}\), where \(C_{LM}\) denotes the field of constants of LM. The following theorem is proved:
Let K be an algebraically closed ordinary differential field of characteristic zero. Let R be a differential field extension of K generated by a single element which is differentially algebraic over K. Then the following are equivalent:
(i) \(C_ R=C_ K\) and R depends rationally on arbitrary constants;
(ii) there exists a strongly normal extension of K which contains R.
Reviewer: E.V.Pankrat’ev

MSC:

12H05 Differential algebra
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References:

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[10] DOI: 10.2969/jmsj/03130553
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