Difference Algebra

In what follows we keep the basic notation and conventions of Chapter 1. In particular, by a ring we always mean an associative ring with unity, every ring homomorphism is unitary (maps unity onto unity), every subring of a ring contains the unity of the ring. Unless otherwise indicated, by the module over a ring A we mean a left A-module. Every module over a ring is unitary and every algebra over a commutative ring is also unitary.

In this section we start the discussion of varieties of ordinary difference polynomials. In what follows we use the terminology introduced for the ordinary case in Section 4.5. The whole section, as well as Sections 7.2 and 7.4 – 7.6, is based on the results of R. Cohn. Some of them were obtained in [28], but the main part of the theory first appeared in [41].

In this chapter we consider some basic aspects of the difference Galois theory. The first section is devoted to the study of Galois groups of normal and separable (but not necessarily finite) difference field extensions and the application of the results this study to the problems of compatibility and monadicity. The corresponding theory was developed by P. Evanovich in [60]. The other two sections of the chapter present a review of fundamentals of two approaches to the Picard-Vessiot theory of ordinary difference field extensions. The original version of this theory, which adjusts the main ideas of the the Picard-Vessiot theory of differential fields to difference case, is due to C. Franke. In section 8.2 we give a review of this approach omitting proofs of most of the results (we prove just a few fundamental statements on Picard-Vessiot extensions). The detail exposition of the theory can be found in the fundamental C. Franke’s work [67], in his further papers [68]–[73], and also in the works by R. Infante [88] and [90]. Section 8.3 provides a review of the basics of a Galois theory of difference equations developed by M. Singer and M. van der Put. This theory, which is based on the study of Picard-Vessiot difference rings, is perfectly presented in the monograph [159] where the reader can also find interesting applications of the results on difference Galois groups to the analytic theory of difference equations.