安德烈-丘德诺夫斯基-卡茨的理论
Le théorème d'André-Chudnovsky-Katz
摘要：
本硕士二年级论文的主题是研究安德烈-丘德诺夫斯基-卡茨定理的结构的非零微分方程的解决方案的最低阶与系数的$\overline{\mathbb{Q}}(z)$满足的$G$ -功能。我们首先介绍了整体幂零微分算子的理论，其主要结果是Katz定理，它指出，他们是fuchsian与有理指数。然后，我们给出了充分的证明Chudnovsky定理意味着最小的非零微分算子的系数在$\overline{\mathbb{Q}}(z)$的$G$ -函数满足一个适度的增长条件的一些算子称为Galochkin条件。最后，我们概述的证明安德烈-Bjueri定理建立之间的等价性Galochkin条件和Bjueri条件，这意味着全球的零。这使我们能够证明安德烈-丘德诺夫斯基-卡茨定理的要点。
英文摘要：
The subject of this Master 2 thesis is the study of the André-Chudnovsky-Katz theorem on the structure of the solution of the nonzero differential equation of minimal order with coefficients in $\overline{\mathbb{Q}}(z)$ satisfied by a $G$-function. We begin by presenting the theory of globally nilpotent differential operators, of which the main result is the Katz theorem, which states that they are fuchsian with rational exponents. We then give a full proof of the Chudnovsky theorem implying that the minimal nonzero differential operator with coefficients in $\overline{\mathbb{Q}}(z)$ of a $G$-function satisfies a moderate growth condition on some denominators called the Galochkin condition. We finally outline the proof of the André-Bombieri theorem establishing the equivalence between the Galochkin condition and the Bombieri condition, which implies the global nilpotence. This allows us to prove the main point of the André-Chudnovsky-Katz theorem.
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