2016/06/27 by James Freitag, Freitag, James, Rahim Moosa +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #03C60 #12H05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Logic (math.LO) #Nonlinear Waves and Solitons #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1606.08492
openalex publication_date 2016/06/27 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Hrushovski's generalization and application of [Jouanolou, "Hypersurfaces\nsolutions d'une 'equation de Pfaff analytique", Mathematische Annalen, 232\n(3):239--245, 1978] is here refined and extended to the partial differential\nsetting with possibly nonconstant coefficient fields. In particular, it is\nshown that if X is a differential-algebraic variety over a partial\ndifferential field F that is finitely generated over its constant field F0,\nthen there exists a dominant differential-rational map from X to the constant\npoints of an algebraic variety V over F0, such that all but finitely many\ncodimension one subvarieties of X over F arise as pull-backs of algebraic\nsubvarieties of V over F0. As an application, it is shown that the algebraic\nsolutions to a first order algebraic differential equation over C(t) are of\nbounded height, answering a question of Eremenko. Two expected model-theoretic\napplications to DCF0,m are also given: 1) Lascar rank and Morley rank agree\nin dimension two, and 2) dimension one strongly minimal sets orthogonal to the\nconstants are \ℵ0-categorical. A detailed exposition of Hrushovski's\noriginal (unpublished) theorem is included, influenced by [Ghys, " `A propos\nd'un th 'eor `eme de J.-P. Jouanolou concernant les feuilles ferm 'ees des\nfeuilletages holomorphes", Rend. Circ. Mat. Palermo (2), 49(1):175--180, 2000.\n