Differential Galois theory and linear algebraic groups over algebraic function fields

微分伽罗瓦理论给每一个线性微分方程分配一个矩阵群，称为微分伽罗瓦群。微分伽罗瓦群包含由方程的解产生的微分场的信息。它还测量了解之间的代数关系。一类常线性微分方程的微分伽罗瓦群是一类线性代数群。微分反伽罗瓦问题是指在给定的微分域上，哪些线性代数群以微分伽罗瓦群的形式出现。在有理函数域k(x)上，在代数闭域k上，我们已经知道了近20年来每一个线性代数群都会出现。项目的第一部分是证明k(x)的绝对微分伽罗瓦群是一个自由的原代数群。该方法基于场补片方法和对微分嵌入问题的研究。对于依赖于附加离散参数的微分方程，可以定义它的σ -微分伽罗瓦群，它是微分伽罗瓦群的一个子群。它可以被看作是后者的改进版本，因为它测量解之间的差分代数关系。微分伽罗瓦群是线性微分代数群。在这个项目的第二部分，我们将研究C(x)上相应的逆问题。本课题的第三部分致力于线性代数群下的环量问题的研究：利用域补码方法，我们旨在证明约简范数的一个局部-全局原理。

Differential Galois theory assigns a matrix group, called differential Galois group, to each linear differential equation. The differential Galois group contains information on the differential field generated by the solutions of the equation. It also measures the algebraic relations among the solutions. The differential Galois group of an ordinary linear differential equation is a linear algebraic group. The inverse differential Galois problem asks which linear algebraic groups occur as differential Galois groups over a given differential field. Over rational function fields k(x) over algebraically closed fields k it has been known for almost 20 years that every linear algebraic group occurs. The first part of the project is to show that in addition, the absolute differential Galois group of k(x) is a free proalgebraic group. The approach is based on field patching methods and on the study of differential embedding problems.For a differential equation depending on an additional discrete parameter, one can define its sigma-differential Galois group, which is a subgroup of the differential Galois group. It can be regarded as a refined version of the latter, as it measures difference-algebraic relations among the solutions. Sigma-differential Galois groups are linear differential algebraic groups. In the second part of this project, we will study the corresponding inverse problem over C(x).The third part of this project is devoted to the study of torsors under linear algebraic groups: Using the field patching method, we aim to prove a local-global principle for reduced norms.