The Alexander polynomial of a plane curve singularity via the ring of functions on it

We prove two formulae that express the Alexander polynomial C of several variables of a plane curve singularity C in terms of the ring O C of germs of analytic functions on the curve. One of them expresses C in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring O C. The other one gives the coefficients of the Alexander polynomial C as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).