Some Properties of Coefficients of the Kolchin Dimension Polynomial

Abstract

This paper presents a formula expressing Macaulay constants of a numerical polynomial through its minimizing coefficients. From this, we obtain that Macaulay constants of Kolchin dimension polynomials do not decrease. For the minimal differential dimension polynomial $$ {\omega}_{\mathcal{G}/\mathcal{F}} $$ (this concept was introduced by W. Sitt) we will prove a criterion for Macaulay constants to be equal. In this case, as our example shows, there are no bounds from above to the Macaulay constants of the polynomial $$ {\omega}_{\xi /\mathcal{F}} $$ for $$ \mathcal{G}=\mathcal{F}\left\langle \xi \right\rangle . $$