Adapting Rabin's theorem for differential fields

Authors:
Russell Miller;Alexey Ovchinnikov
Affiliations:
Department of Mathematics, Queens College, City University of New York, Flushing, NY and Department of Mathematics, CUNY Graduate Center, New York, NY;Department of Mathematics, Queens College, City University of New York, Flushing, NY
Venue:
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
Year:
2011

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CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe

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Abstract

Harrington extended the first half of Rabin's Theorem to differential fields, proving that every computable differential field can be viewed as a computably enumerable subfield of a computable presentation of its differential closure. For fields F, the second half of Rabin's Theorem says that this subfield is Turing-equivalent to the set of irreducible polynomials in F[X]. We investigate possible extensions of this second half, asking both about the degree of the differential field K within its differential closure and about the degree of the set of constraints for K, which forms the closest analogue to the set of irreducible polynomials.