Finiteness theorems on hypersurfaces in partial differential-algebraic geometry

Abstract

Hrushovski's generalization and application of Jouanolou (1978) [9] 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 F0, then there exists a dominant differential-rational map from X to the constant points of an algebraic variety V over F0, such that all but finitely many codimension one subvarieties of X over F arise as pull-backs of algebraic subvarieties of V over F0. 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 DCF0,m 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 ℵ0-categorical. A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by Ghys (2000) [5].