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We introduce a new type of reduction of inversive difference polynomials thatis associated with a partition of the basic set of automorphisms $\sigma$ anduses a generalization of the concept of effective order of a differencepolynomial. Then we develop the corresponding method of characteristic sets andapply it to prove the existence and obtain a method of computation ofmultivariate dimension polynomials of a new type that describe thetranscendence degrees of intermediate fields of finitely generated inversivedifference field extensions obtained by adjoining transforms of the generatorswhose orders with respect to the components of the partition of $\sigma$ arebounded by two sequences of natural numbers. We show that such dimensionpolynomials carry essentially more invariants (that is, characteristics of theextension that do not depend on the set of its difference generators) thanstandard (univariate) difference dimension polynomials. We also show how theobtained results can be applied to the equivalence problem for systems ofalgebraic difference equations.

New Multivariate Dimension Polynomials of Inversive Difference Field Extensions