Iterative differential Galois theory: a model theoretic approach

An invitation to model-theoretic Galois theory

We carry out some of Galois's work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions. This exposition of a special case of Poizat's "Une théorie de Galois imaginaire." (1983) has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphi…

Model theoretic dynamics in Galois fashion

We investigate existentially closed models (of a quite arbitrary theory) equipped which an action of a fixed group G. We embed these structures in a monster model D of some well-rounded theory and describe them as PAC substructures of D. Assuming that the class of these existentially closed models is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas - similar to ACFA. Moreover, it codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imagina…

微分伽罗华理论中的参数化D- torsors

Parameterized D-torsors in differential Galois theory

本文研究了参数化D-torsors在微分伽罗华理论中的应用，提出了参数化版本的Kolchin定理，并探讨了广义强正常扩展与log-微分方程之间的关系。

利用伽罗瓦理论分裂微分方程

Splitting differential equations using Galois theory

本文针对代数常微分方程对数导数拉回的内性判定问题，借助模型论伽罗瓦理论与微分代数建立拉回内性和伽罗瓦群短正合列分裂的关联，再用代数群理论得到内性与非内性结果。

关于全超越环境中的可定义伽罗瓦理论与可定义伽罗瓦上同调

On definable Galois theory and definable Galois cohomology in the totally transcendental setting

本文在全超越一阶理论中建立可定义伽罗瓦理论与上同调的联系，给出微分伽罗瓦上同调的余极限公式，并引入微分域和参数集的有界性新概念。

More on Galois cohomology, definability and differential algebraic groups

We make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired on Serre's algebraic twisting) to describe arbitrary fibres in cohomology sequences -- yielding a useful finiteness result on cohomology sets. Applied to the special case of differential fields and Kolchin's constrained cohomology, we prove that the first constrained cohomology set of a differen…