surfacesignature.lib

SINGULAR Library surfacesignature.lib: signature of surface singularity. A library for computing the signature of irreducible surface singularity. The signature of a surface singularity is defined in [3]. The algorithm we use has been proposed in [9]. Let g in C[x,y] define an isolated curve singularity at 0 in C^2 and f:=z^N+g(x,y). The zero-set V:=V(f) in C^3 of f has an isolated singularity at 0. For a small e>0 let V_e:=V(f-e) in C^3 be the Milnor fibre of (V,0) and s: H_2(V_e,R) x H_2(V_e,R) ---> R be the intersection form (cf. [1],[7]). H_2(V_e,R) is an m-dimensional R-vector space, m the Milnor number of (V,0) (cf. [1],[4],[5],[6]), and s is a symmetric bilinear form. Let sigma(f) be the signature of s, called the signature of the surface singularity (V,0). Formulaes to compute the signature are given by Nemethi (cf. [8],[9]) and van Doorn, Steenbrink (cf. [2]). We have implemented three approaches using Puiseux expansions, the resolution of singularities resp. the spectral pairs of the singularity.

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