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  1. 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.
  2. Free, publicly-accessible full text available October 1, 2022
  3. Let K be an inversive difference-differential field and L a (not necessarily inversive) finitely generated difference-differential field extension of K. We consider the natural filtration of the extension L/K associated with a finite system \eta of its difference-differential generators and prove that for any intermediate difference-differential 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 Hilbert-type 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 somemore »properties of polynomials \chi_{K, F,\eta}(t) and use them for the study of the Krull-type 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.« less