Gröbner bases with respect to several term orderings and multivariate dimension polynomials

ABSTRACT

Let D be a ring of Ore polynomials in m variables x 1 ,...,x m over a field K and let a partition of the set {x 1 ,...,x m}into p disjoint subsets be fixed, so that D can be treated as a filtered ring with the natural p-dimensional ltration associated with the partition.We introduce a special type of reduction in a finitely generated free D-module and develop the corresponding Gröbner basis technique that allows one to prove the existence and find invariants of a dimension polynomial in p variables associated with a finitely generated D-module. We also outline a method of computation of such a polynomial and obtain an essential generalization of the Kolchin theorem on the dimension polynomial of a differential field extension.

References

1. Buchberger, B. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph. Thesis. Univ. of Innsbruck, Austria, 1965.
2. Johnson, J. Differential Dimension Polynomials and a Fundamental Theorem on Differential Modules. Amer. J. Math., 91 (1969), 239--248.
3. Kolchin, E.R. The notion of dimension in the theory of algebraic differential equations. Bull.Amer. Math. Soc., 70 (1964), 570--573.
4. Pankratiev, E.V. Differential and Difference Dimension Polynomials. Kluwer Academic Publishers., Dordrecht, 1998.
5. Levin, A.B. Reduced Grobner bases, free difference-differential modules and difference-differential dimension polynomials. J. Symbolic Comput., 29 (2000), 1--26.

Index Terms

• Computing methodologies

  • Symbolic and algebraic manipulation