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

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.

2-Dimension Polynomial Fitting for the Edge Detection

The gray-scale digital image is two-dimension, most of the previous polynomial fitting methods for edge detection belong to one-dimension methods. The new method of two-dimension polynomial fitting for edge detection is presented. The grey level data of the interest area around the edge in the image are fitted by the two-dimension polynomial function. The edge of interest is identified by finding the maximum of the form of gradient of the fitting function. Because the two-dimension fitting is actually more suitable for the two-dimension image, the fitting results of two dimension method are significantly better than that of the one-dimension method. It is shown through the analysis of the synthesis image that the results of surface fitting and edge identification used of the proposed method are quite good.

Multivariable dimension polynomials and new invariants of differential field extensions

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.

Characteristic polynomials of finitely generated modules over Weyl algebras

In this paper we modify the classical Gröbner basis technique and prove the existence of a characteristic polynomial in two variables associated with a finitely generated module over a Weyl algebra. We determine invariants of such a polynomial and show that some of the invariants are not carried by the Bernstein dimension polynomial of the module.

ON PLANARITY OF GRAPHS ON WEYL GROUPS

A graph is constructed whose vertices are elements of a Weyl group and the edges are defined through nonvanishing of Wey!'s dimension polynomial at the point associated with two elements of the Weyl group. We study the planarity of such graphs on Weyl groups whose associated root system is irreducible. These graphs include four families of infinite number of graphs. We show that very few graphs, essentially five of them, are planar.

COMPUTATION OF DIMENSION POLYNOMIALS

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.