The Differential Dimension Polynomial for Characterizable Differential Ideals

We introduce a special type of reduction in the ring of differential polynomials and develop the appropriate technique of characteristic sets that allows to generalize the classical Kolchin's theorem on differential dimension polynomial and find new differential birational invariants of a finitely generated differential field extension.

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COMPUTATION OF DIMENSION POLYNOMIALS

International Journal of Algebra and Computation ◽

10.1142/s0218196792000098 ◽

1992 ◽

Vol 02 (02) ◽

pp. 117-137 ◽

Cited By ~ 14

Author(s):

M.V. KONDRAT’EVA ◽

A.B. LEVIN ◽

A.V. MIKHALEV ◽

E.V. PANKRAT’EV

Keyword(s):

Differential Dimension ◽

Arbitrary Partition ◽

Dimension Polynomial

The consideration of differential versions of Hilbert dimension polynomials is due to A. Einstein [1] and E. Kolchin [2] (one can find the coverage of the theory of differential dimension polynomials in [6]). In this paper we introduce the notion of a dimension polynomial of a subset of ℕm associated with arbitrary partition of the set {1,…, m} into disjoint nonempty subsets (m∈ℕ, ℕ denoting the set of all nonnegative integers). The theory of such polynomials is developed. The importance of our considerations is connected with the fact that the computation of differential and difference dimen sion polynomials may be reduced to the computation of some dimension polynomials of subsets of ℕm where m∈ℕ (see [3, p. 115], [5]). We also give some methods and algorithms for computation of dimension polynomials.

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Estimates for the coefficients of differential dimension polynomials

Mathematics of Computation ◽

10.1090/mcom/3429 ◽

2019 ◽

Vol 88 (320) ◽

pp. 2959-2985 ◽

Cited By ~ 2

Author(s):

Omar León Sánchez

Keyword(s):

Differential Dimension

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HEIGHT PLUS DIFFERENTIAL DIMENSION IN COMMUTATIVE NOETHERIAN RINGS

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-30.1.15 ◽

1984 ◽

Vol s2-30 (1) ◽

pp. 15-20 ◽

Cited By ~ 2

Author(s):

K. R. GOODEARL ◽

T. H. LENAGAN ◽

P. C. ROBERTS

Keyword(s):

Noetherian Rings ◽

Differential Dimension

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BIVARIATE DIMENSION POLYNOMIALS AND NEW INVARIANTS OF FINITELY GENERATED D-MODULES

International Journal of Algebra and Computation ◽

10.1142/s0218196713500409 ◽

2013 ◽

Vol 23 (07) ◽

pp. 1625-1651 ◽

Cited By ~ 1

Author(s):

CHRISTIAN DÖNCH ◽

ALEXANDER LEVIN

Keyword(s):

Gröbner Basis ◽

Weyl Algebra ◽

Isomorphism Problem ◽

Grobner Basis ◽

Finitely Generated ◽

Finitely Generated Module ◽

D Modules ◽

Dimension Polynomial

In this paper, we generalize the classical Gröbner basis technique to prove the existence and present a method of computation of a dimension polynomial in two variables associated with a finitely generated D-module, that is, a finitely generated module over a Weyl algebra. We also present corresponding algorithms and examples of computation of such polynomials and show that bivariate dimension polynomials contain invariants of a D-module, which are not carried by its Bernstein dimension polynomial. Then we apply the obtained results to the isomorphism problem for D-modules.

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