  We prove two formulae which express the Alexander polynomial $\Delta^C$ of
several variables of a plane curve singularity $C$ in terms of the ring ${\cal
O}_{C}$ of germs of analytic functions on the curve. One of them expresses
$\Delta^C$ in terms of dimensions of some factors corresponding to a
(multi-indexed) filtration on the ring ${\cal O}_{C}$. The other one gives the
coefficients of the Alexander polynomial $\Delta^C$ as Euler characteristics of
some explicitly described spaces (complements to arrangements of projective
hyperplanes). The final version of this article will be published in the Duke
Mathematical Journal.
