Picard-Vessiot Theory and Ziglin′s Theorem

A Categorical Approach to Picard-vessiot Theory

Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems on existence and uniqueness of Picard-Vessiot rings, as well as on Galois groups corresponding to the Picard-Vessiot rings. As the present approach restrict...

Relative invariants, difference equations, and the Picard-Vessiot theory

Acknowledgements First of all I would like to thank my supervisor Professor T. Kimura. He taught me how to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master's thesis. I can not give enough thanks to his heartwarming encouragement all...

Quasialgebraicity of Picard–vessiot Fields

We prove that under certain spectral assumptions on the monodromy group, solutions of the Fuchsian system of linear equations on the Riemann sphere admit explicit global bounds on the number of their isolated zeros.

The Picard–vessiot Antiderivative Closure

F is a differential field of characteristic zero with algebraically closed field of constants C. A Picard–Vessiot antiderivative closure of F is a differential field extension E ⊃ F which is a union of Picard–Vessiot extensions of F , each obtained by iterated adjunction of antiderivatives, and such that every such Picard– Vessiot extension of F has an isomorphic copy in E. The group G of diffe...

The Picard-Vessiot Theory of Homogeneous Linear Ordinary Differential Equations.