%ALevin, Alexander%Anull Ed.%BJournal Name: Lecture notes in computer science; Journal Volume: 11989 %D2019%I %JJournal Name: Lecture notes in computer science; Journal Volume: 11989 %K %MOSTI ID: 10301698 %PMedium: X %THilbert-Type Dimension Polynomials of Intermediate Difference-Differential Field Extensions %XLet 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 some 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. %0Journal Article