Bounds for ${e}_{0}(\mathbb{M})$ and ${e}_{1}(\mathbb{M})$

Abstract

In this chapter we prove lower and upper bounds for the first two coefficients of the Hilbert polynomial which we defined in Chap. 1. We recall that ${e}_{0}(\mathbb{M})$ depends only on q and M, but it does not depend on the good q-filtration $\mathbb{M}$. In contrast ${e}_{1}(\mathbb{M})$ does depend on the filtration $\mathbb{M}$. It is called by Vasconcelos tracking number for its tag position among the different filtrations having the same multiplicity. The coefficient ${e}_{1}(\mathbb{M})$ is also called the Chern number (see [111]).