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Preprint Number 1062

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1062. James Freitag and Rahim Moosa
Finiteness theorems on hypersurfaces in partial differential-algebraic geometry

Submission date: 29 June 2016


Hrushovski's generalization and application of Jouanolou's work on foliations is here refined and extended to the partial differential setting with possibly nonconstant coefficient fields. In particular, it is shown that if X is a differential-algebraic variety over a partial differential field F that is finitely generated over its constant field F_0, then there exists a dominant differential-rational map from X to the constant points of an algebraic variety V over F_0, such that all but finitely many codimension one subvarieties of X over F arise as pull-backs of algebraic subvarieties of V over F_0. As an application, it is shown that the algebraic solutions to a first order algebraic differential equation over C(t) are of bounded height, answering a question of Eremenko. Two expected model-theoretic applications to differentially closed fields are also given: 1) Lascar rank and Morley rank agree in dimension two, and 2) dimension one strongly minimal sets orthogonal to the constants are ℵ_0-categorical.

Mathematics Subject Classification: 03C60, 12H05

Keywords and phrases: Differential algebra, hypersurfaces, categoricity, heights in function fields

Full text arXiv 1606.08492: pdf, ps.

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