Reduced Gröbner bases, free difference-differential modules and difference-differential dimension polynomials
Difference dimension polynomials play the same role in difference algebra as Hilbert polynomials in commutative algebra or differential dimension polynomials in differential algebra. The paper extends the notion of difference dimension polynomials and differential dimension polynomials introducing the notion of difference-differential polynomials associated with finitely generated difference-differential field extensions. In order to study these dimension polynomials the author defines a notion of reduction in free modules over rings of difference-differential operators which allows to develop an analog of Gröbner basis theory. One of the main theorems of the paper generalizes both Kolchin's theorem on dimension polynomials of differential field extensions and the corresponding theorem for difference fields due to the author. The developed theory is then applied to get a method for computing the dimension polynomial associated with a system of linear difference-differential equations.
difference-differential dimension polynomials
difference-differential field extensions
scientific article; zbMATH DE number 1538749
scientific article