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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 Kahler 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.

Let K be an inversive differencedifferential field and L a (not necessarily inversive) finitely generated differencedifferential field extension of K. We consider the natural filtration of the extension L/K associated with a finite system \eta of its differencedifferential generators and prove that for any intermediate differencedifferential field F, the transcendence degrees of the components of the induced filtration of F are expressed by a certain numerical polynomial \chi_{K, F,\eta}(t). This polynomial is closely connected with the dimension Hilberttype polynomial of a submodule of the module of K\"ahler differentials $\Omega_{L^{\ast}K} where L^{\ast} is the inversive closure of L. We prove some properties of polynomials \chi_{K, F,\eta}(t) and use them for the study of the Krulltype dimension of the extension L/K. In the last part of the paper, we present a generalization of the obtained results to multidimensional filtrations of L/K associated with partitions of the sets of basic derivations and translations.

We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate dimension polynomial associated with a nonreflexive differencedifferential ideal in the algebra of differencedifferential polynomials with several basic derivations and one translation. As a consequence, we obtain a new proof and a method of computation of the dimension polynomial of a nonreflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. We also discuss applications of our results to systems of algebraic differencedifferential equations.

Multivariate dimension polynomials associated with finitely generated differential and difference field extensions arise as natural generalizations of the univariate differential and difference dimension polynomials. It turns out, however, that they carry more information about the corresponding extensions than their univariate counterparts. We extend the known results on multivariate dimension polynomials to the case of differencedifferential field extensions with arbitrary partitions of sets of basic operators. We also describe some properties of multivariate dimension polynomials and their invariants.

We present a difference algebraic technique for the evaluation of the Einstein's strength of quasilinear partial difference equations and some systems of such equations. Our approach is based on the properties of difference dimension polynomials that express the Einstein's strength and on the characteristic set method for computing such polynomials. The obtained results are applied to the comparative analysis of difference schemes for some chemical reactiondiffusion equations.

We consider Hilberttype functions associated with finitely generated inversive difference field extensions and systems of algebraic difference equations in the case when the translations are assigned positive integer weights. We prove that such functions are quasipolynomials that can be represented as alternating sums of Ehrhart quasipolynomials of rational conic polytopes. In particular, we generalize the author's results on difference dimension polynomials and their invariants to the case of inversive difference fields with weighted basic automorphisms.