Reduced Gröbner Bases, Free Difference–Differential Modules and Difference–Differential Dimension Polynomials
✍ Scribed by Alexander Levin
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 442 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0747-7171
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✦ Synopsis
We define a special type of reduction in a free left module over a ring of differencedifferential 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 differencedifferential module equipped with the natural double filtration. The results obtained are applied to the study of difference-differential field extensions and systems of differencedifferential 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.