Graded Betti Numbers of the Logarithmic Derivation Module

Abstract

Let $Q\in \K[x_1,...,x_n] = S$ be a homogeneous polynomial of degree $d$. The freeness of the logarithmic derivation module, $D(Q)$, and of its natural generalizations, has been widely studied. In the free case, $D(Q)\simeq {⨁}_{i=1}^{n}S(-d_i)$ where the $d_i$'s are the exponents of the module; and as a direct consequence of the Saito-Ziegler criterion, the formula $d={\sum }_i{d}_i$ holds. In this paper we give a generalization of this formula in the non-free case. Moreover, we show that an equivalent formula is also true in the quasi-homogeneous case, and show to what extent it can be generalized for arbitrary polynomials.