Finiteness theorems on hypersurfaces in partial differential-algebraic geometry

TL;DR

This paper extends finiteness theorems to hypersurfaces in partial differential-algebraic geometry, showing boundedness of algebraic solutions and exploring model-theoretic properties of differential fields.

Contribution

It refines Hrushovski's theorem to the partial differential setting with nonconstant coefficients and applies it to algebraic solutions and model theory.

Findings

Algebraic solutions to certain differential equations have bounded height.

Lascar rank and Morley rank coincide in dimension two.

Strongly minimal sets orthogonal to constants are -categorical.

Abstract

Hrushovski's generalization and application of [Jouanolou, "Hypersurfaces solutions d'une \'equation 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 A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by [Ghys, "\`A propos d'un th\'eor\`eme de J.-P. Jouanolou concernant les feuilles ferm\'ees des feuilletages holomorphes", Rend. Circ. Mat. Palermo (2)}, 49(1):175--180, 2000.