Dimension Polynomials of Intermediate Fields of Inversive Difference Field Extensions
Let K be an inversive difference field, L a finitely generated inversive difference field extension of K, and F an intermediate inversive difference field of this extension. We prove the existence and establish properties and invariants of a numerical polynomial that describes the filtration of F induced by the natural filtration of the extension L/K associated with its generators. Then we introduce concepts of type and dimension of the extension L/K considering chains of its intermediate fields. Using properties of dimension polynomials of intermediate fields we obtain relationships between the type and dimension of L/K and difference birational invariants of this extension carried by its dimension polynomials. Finally, we present a generalization of the obtained results to the case of multivariate dimension polynomials associated with a given inversive difference field extension and a partition of the basic set of translations.
KeywordsInversive difference field Inversive difference module Filtration Dimension polynomial
- 1.Einstein, A.: The Meaning of Relativity. Appendix II (Generalization of gravitation theory), 4th edn., pp. 133–165. Princeton (1953)Google Scholar
- 6.Kolchin, E.R.: Some problems in differential algebra. In: Proceedings of the International Congress of Mathematicians (Moscow - 1966), Moscow, pp. 269–276 (1968)Google Scholar
- 12.Levin, A.B.: Computation of the strength of systems of difference equations via generalized Gröbner bases. In: Grobner Bases in Symbolic Analysis, pp. 43–73. Walter de Gruyter (2007)Google Scholar
- 17.Mikhalev, A.V., Pankratev, E.V.: Computer Algebra. Calculations in Differential and Difference Algebra. Moscow State Univ. Press, Moscow (1989)Google Scholar