Let K be a partial differential field of characteristic zero with m commuting derivations, and let \(\mathcal {D}\) be the corresponding ring of differential operators. For a finitely generated left \(\mathcal {D}\)-module M with a fixed finite generating set, we study order-window vector subspaces Mr, s generated by derivatives of generators whose total order lies between s and r (r > s).

We introduce a reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms and uses a generalization of the concept of effective order of a difference polynomial. Then we develop the corresponding method of characteristic sets and apply it to prove the existence and obtain a method of computation of multivariate dimension polynomials that describe the transcendence degrees of intermediate fields of finitely generated inversive difference field extensions obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of the basic set are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on its difference generators) than standard (univariate) difference dimension polynomials. We also show how the obtained results can be applied to the equivalence problem for systems of algebraic difference equations.

We introduce a new type of reduction in a free difference module over a difference field that uses a generalization of the concept of effective order of a difference polynomial. Then we define the concept of a generalized characteristic set of such a module, establish some properties of these characteristic sets and use them to prove the existence, outline a method of computation and find invariants of a dimension polynomial in two variables associated with a finitely generated difference module. As a consequence of these results, we obtain a new type of bivariate dimension polynomials of finitely generated difference field extensions. We also explain the relationship between these dimension polynomials and the concept of Einstein's strength of a system of difference equations.

We present a method of Gröbner bases with respect to several term orderings and use it to obtain new results on multivariate dimension polynomials of inversive difference modules. Then we use the difference structure of the module of Kähler differentials associated with a finitely generated inversive difference field extension of a given difference transcendence degree to describe the form of a multivariate difference dimension polynomial of the extension.

In this paper we obtain new results on multivariate dimension polynomials of differential field extensions associated with partitions of basic sets of derivations. We prove that the coefficient of the summand of the highest possible degree in the canonical representation of such a polynomial is equal to the differential transcendence degree of the extension. We also give necessary and sufficient conditions under which the multivariate dimension polynomial of a differential field extension of a given differential transcendence degree has the simplest possible form. Furthermore, we describe some relationships between a multivariate dimension polynomial of a differential field extension and dimensional characteristics of subextensions defined by subsets of the basic sets of derivations.