Some Properties of Coefficients Kolchin Dimension Polynomial

TL;DR

This paper investigates properties of Kolchin dimension polynomials, providing formulas for Macaulay constants, proving their non-decreasing nature, and establishing criteria for their equality in minimal differential dimension polynomials.

Contribution

It introduces a formula linking Macaulay constants to minimizing coefficients and characterizes when these constants are equal in minimal differential dimension polynomials.

Findings

Macaulay constants of Kolchin dimension polynomials do not decrease.

A criterion for equality of Macaulay constants in minimal differential dimension polynomials.

No upper bounds for Macaulay constants of the dimension polynomial for the starting generator.

Abstract

The article presents a formula expressing Macaulay constants of a numerical polynomial through its minimizing coefficients. From this, we have that Macaulay constants of Kolchin dimension polynomials do not decrease. For the minimal differential dimension polynomial (this concept was introduced by W.Sitt in [5]) we will prove a criterion for Macaulay constants to be equal. In this case, as the example (2) shows, there are no bounds from above to the Macaulay constants of the dimension polynomial for starting generator.