Some properties of coefficients of the Kolchin dimension polynomial

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 $\mathcal G=\mathcal F\langle\xi\rangle$.