Submission date: 29 June 2016

Abstract:

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.