  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, $D(Q) \simeq
\bigoplus_{i=1}^n S(-d_i)$ 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.
