Paper detail

Finiteness theorems on hypersurfaces in partial differential-algebraic geometry

Hrushovski's generalization and application of [Jouanolou, "Hypersurfaces solutions d'une équation de Pfaff analytique", Mathematische Annalen, 232 (3):239--245, 1978] 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 DCF_{0,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 \aleph_0-categorical. A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by [Ghys, "À propos d'un théorème de J.-P. Jouanolou concernant les feuilles fermées des feuilletages holomorphes", Rend. Circ. Mat. Palermo (2)}, 49(1):175--180, 2000.