arXiv subjects
Publications and source records attributed to Katsutoshi Amano.
There is a serious discrepancy among literature on the Picard-Vessiot theory in positive characteristics (for iterative differential fields). It is about descriptions of Galois correspondence. We should use affine group schemes instead of algebraic matrix groups to obtain a suitable Galois correspondence. If one adheres to algebraic matrix groups, then his correspondence may not be bijective.
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.