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

A. Campillo·F. Delgado·S. M. Gusein-Zade
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Abstract

We give two formulae which express the Alexander polynomial $Δ^C$ 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 $Δ^C$ in terms of dimensions of some factorspaces corresponding to a (multi-indexed) filtration on the ring ${\cal O}_{C}$. The other one gives the coefficients of the Alexander polynomial $Δ^C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).

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A. Campillo, F. Delgado, S. M. Gusein-Zade. 2000-02-07. The Alexander polynomial of a plane curve singularity and the ring of functions on it. https://arxiv.org/abs/math/0002052

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