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We prove two formulae which 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).

It was shown that the Alexander polynomial (of several variables) of a (reducible) plane curve singularity coincides with the (generalized) Poincaré polynomial of the multi-indexed filtration defined by the curve on the ring [Formula: see text] of germs of functions of two variables. The initial proof of the result was rather complicated (it used analytical, topological and combinatorial arguments). Here we give a new proof based on the notion of the integral with respect to the Euler characteristic over the projectivization of the space [Formula: see text] — the notion similar to (and inspired by) the notion of the motivic integration.

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