On a general difference Galois theory II

Difference Nullstellensatz in the case of finite group

“…A difference algebraic analogue is introduced in Section 3.3 and is called the ring of functions. Such a construction appeared in many papers, for example [16,10,11].…”

Can we quantize Galois theory ?

“…and the difference equation ( 8) reduces to hypergeometric equation (7). In this sense the Heine series is a q-analogue of the hypergeometric series.…”

“…that is a difference algebra morphism so that compatible with difference operation σ and Σ. General difference Galois theory is established by Shuji Morikawa [7]. The theory depends on the same idea as the general differential theory.…”

Hiroshi Umemura et les mathématiques françaises

“…Malheureusement certaines définitions et les démonstrations sont manquantes. Umemura s'est intéressé dans [83] (avec Morikawa) à l'application de sa théorie à l'intégrabilité des systèmes dynamiques. Leur philosophie est : « In theory of dynamical systems, we believe in general what is integrable is abelian ».…”