Finiteness theorems on hypersurfaces in partial differential-algebraic geometry
Finiteness theorems on hypersurfaces in partial differential-algebraic geometry
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DOI：
10.1016/j.aim.2017.04.008
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发表时间：
2016-06
期刊：
影响因子：
--
通讯作者：
J. Freitag;Rahim Moosa
中科院分区：
文献类型：
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作者：
J. Freitag;Rahim Moosa
Hrushovski 对 Jouanolou (1978)[9] 的概括和应用在这里被细化并扩展到具有可能非恒定系数场的偏微分设置。特别地，它表明，如果 X 是偏微分域 F 上的微分代数簇，而该偏微分域 F 是在其常数域 F 0 上有限生成的，则存在从 X 到 F 0 上代数簇 V 的常数点的显性微分有理映射，使得除了 F 上 X 的有限多个余维一子簇之外，所有 X 上的余维一子簇都作为 V 上 F 0 上的代数子簇的回拉而出现。作为一个应用，表明C (t) 上的一阶代数微分方程的代数解具有有界高度，回答了 Eremenko 的问题。还给出了 DCF 0, m 的两个预期模型理论应用：1）Lascar 秩和 Morley 秩在第二维上一致，2）与常数正交的第一维强最小集是 ℵ 0-分类的。受 Ghys (2000)[5] 的影响，包括对 Hrushovski 原始（未发表）定理的详细阐述。
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 A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by Ghys (2000)[5].