### D.6.18 surfacesignature_lib

Library:
  surfacesignature.lib
Purpose:
  signature of surface singularity

Authors:
  Gerhard Pfister pfister@mathematik.uni-kl.de
  Muhammad Ahsan Banyamin ahsanbanyamin@gmail.com
  Stefan Steidel steidel@mathematik.uni-kl.de

Overview:
  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.

References:
  [1] Arnold, V.I.; Gusein-Zade, S.M.; Varchenko, A.N.: Singularities of Differentiable Mappings. Vol. 1,2, Birkh"auser (1988). [2] van Doorn, M.G.M.; Steenbrink, J.H.M.: A supplement to the monodromy theorem. Abh. Math. Sem. Univ. Hamburg 59, 225-233 (1989). [3] Durfee, A.H.: The Signature of Smoothings of Complex Surface Singularities. Mathematische Annalen 232, 85-98 (1978). [4] de Jong, T.; Pfister, G.: Local Analytic Geometry. Vieweg (2000). [5] Kerner, D.; Nemethi, A.: The Milnor fibre signature is not semi-continous. arXiv:0907.5252 (2009).
  [6] Kulikov, V.S.: Mixed Hodge Structures and Singularities. Cambridge Tracts in Mathematics 132, Cambridge University Press (1998). [7] Nemethi, A.: The real Seifert form and the spectral pairs of isolated hypersurface singularities. Compositio Mathematica 98, 23-41 (1995). [8] Nemethi, A.: Dedekind sums and the signature of f(x,y)+z^N. Selecta Mathematica, New series, Vol. 4, 361-376 (1998).
  [9] Nemethi, A.: The Signature of f(x,y)+z^$. Proceedings of Real and Complex Singularities (C.T.C. Wall's 60th birthday meeting, Liverpool (England), August 1996), London Math. Soc. Lecture Notes Series 263, 131--149 (1999).

Procedures:

> D.6.18.1 signatureBrieskorn |  | signature of singularity x^a1+y^a2+z^a3
> D.6.18.2 signaturePuiseux |  | signature of singularity z^N+f(x,y)=0, f irred.
> D.6.18.3 signatureNemethi |  | signature of singularity z^N+f(x,y)=0

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