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Finiteness theorems on hypersurfaces in partial differential-algebraic geometry.
- Source :
-
Advances in Mathematics . Jul2017, Vol. 314, p726-755. 30p. - Publication Year :
- 2017
-
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 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 ℵ 0 -categorical. A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by Ghys (2000) [5] . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00018708
- Volume :
- 314
- Database :
- Academic Search Index
- Journal :
- Advances in Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 123428232
- Full Text :
- https://doi.org/10.1016/j.aim.2017.04.008