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.
Categories: math.AG