Reduced Gröbner Bases, Free Difference–Differential Modules and Difference–Differential Dimension Polynomials

We define a special type of reduction in a free left module over a ring of difference–differential operators and use the idea of the Gröbner basis method to develop a technique that allows us to determine the Hilbert function of a finitely generated difference–differential module equipped with the natural double filtration. The results obtained are applied to the study of difference–differential field extensions and systems of difference–differential equations. We prove a theorem on difference–differential dimension polynomial that generalizes both the classical Kolchin’s theorem on dimension polynomial of a differential field extension and the corresponding author’s result for difference fields. We also determine invariants of a difference–differential dimension polynomial and consider a method of computation of the dimension polynomial associated with a system of linear difference–differential equations.