Some Properties of Coefficients of the Kolchin Dimension Polynomial

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