Model Theory of Differential Fields
Model Theory of Differential Fields
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微分场模型论
DOI：
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发表时间：
2000
期刊：
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通讯作者：
D. Marker
中科院分区：
文献类型：
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作者：
D. Marker
本文概述了微分闭合场的模型理论，这是一个有趣的环境，人们可以在其中使用模型理论方法来获得代数信息。文章最后用一个例子说明了如何在丢番图应用程序中使用这些信息。微分场是配备了导数δ：K→K的域K；回想一下，这意味着对于x，y∈K，我们有δ(x+y)=δ(X)+δ(Y)和δ(Xy)=xδ(Y)+yδ(X)。粗略地说，当这样的域包含足够多的常微分方程解时，称为差分闭。这种设置允许人们使用模型理论的方法，特别是维度理论的思想，来获得有趣的代数信息。在这堂课中，我概述了微分闭合场的模型理论，最后以一个例子--Hrushovski证明特征零点中的Mordell-lang猜想--展示了这一领域的模型理论方法如何应用于丢番图。我不会给出主要定理的证明。第1-3节中的大部分材料可在[Marker等人]中找到。1996年]，而第4节中的材料可在[Hrushovski and Sokolovic≥，2001年；Pillay1996]中找到。关于微分代数的主要参考文献是[Kolchin 1973]，尽管非常易读的[Kaplansky 1957]包含了这里所需的大部分基础知识，最近的[Magid 1994]也是如此。这本书还介绍了微分代数及其与丢番图几何的联系。我们建议读者参考这些来源以参考原始文献。1.差分闭合字段在本文中，所有字段都将具有特征零。微分域是具有导数δ：K→K的域K，其常数域是C={x∈K：δ(X)=0}。我们将使用语言L={+，−，·，δ，0，1}来研究微分域，这种语言是由一元函数符号δ扩充的环的语言。的理论。
This article surveys the model theory of differentially closed fields, an interesting setting where one can use model-theoretic methods to obtain algebraic information. The article concludes with one example showing how this information can be used in diophantine applications. A differential field is a field K equipped with a derivation δ : K → K; recall that this means that, for x, y ∈ K, we have δ(x + y) = δ(x) + δ(y) and δ(xy) = x δ(y) + yδ(x). Roughly speaking, such a field is called differentially closed when it contains enough solutions of ordinary differential equations. This setting allows one to use model-theoretic methods, and particularly dimensiontheoretic ideas, to obtain interesting algebraic information. In this lecture I give a survey of the model theory of differentially closed fields, concluding with an example — Hrushovski’s proof of the Mordell–Lang conjecture in characteristic zero — showing how model-theoretic methods in this area can be used in diophantine applications. I will not give the proofs of the main theorems. Most of the material in Sections 1–3 can be found in [Marker et al. 1996], while the material in Section 4 can be found in [Hrushovski and Sokolovic ≥ 2001; Pillay 1996]. The primary reference on differential algebra is [Kolchin 1973], though the very readable [Kaplansky 1957] contains most of the basics needed here, as does the more recent [Magid 1994]. The book [Buium 1994] also contains an introduction to differential algebra and its connections to diophantine geometry. We refer the reader to these sources for references to the original literature. 1. Differentially Closed Fields Throughout this article all fields will have characteristic zero. A differential field is a field K equipped with a derivation δ : K → K. The field of constants is C = {x ∈ K : δ(x) = 0}. We will study differential fields using the language L = {+,− , · , δ, 0, 1}, the language of rings augmented by a unary function symbol δ. The theory of