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

Abstract

This is a compilation of three separated studies. The main part (Part 3) deals with a unified Picard-Vessiot theory including Picard-Vessiot theories of differential and difference equations. Changes: [v1 -> v2] Corrected descriptions on "ZXi-linearly independece" in Part 2 (p11, p13, p16) [v2 -> v3] The proof of Proposition 3.5.7 (iii) in v2 is not completed. (Only the proof of Proposition 3.5.8 (ii) is influenced and a revised proof is given.) [v3 -> v4] Lemma 3.5.10 in the previous version is false (see Remark 3.5.10 in the presented version). The proof of Prop. 3.5.9 (ii) and the proof of Lemma 3.9.9 are influenced and revised proofs are given. There are several minor changes too. [v4 -> v5] Added some explanation before Contents, several minor changes. This shall be the final replacement.