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COMPUTATION OF THE EULER CHARACTERISTIC OF THE MILNOR FIBRE OF PLANE CURVE SINGULARITIES.

Byline: Muhammad Ahsan Binyamin, Rabia, Saba Naz and Sonia Tariq

ABSTRACT

In this article we give the implementation of two algorithms to compute the Euler characteristic of Milnor fibre of reduced plane curve singularities in computer algebra system SINGULAR.

Keywords-: Euler characteristic of Milnor fibre, Hamburger-Noether expansion, Computer algebra system SINGULAR, Irreducible curve singularities.

1. INTRODUCTION

In this short note we want to compare two approaches to compute the Euler characteristic of the Milnor fibre of a plane curve singularity. From the computational point of view, one approach uses the resolution graph and the multiplicity sequence of the curve. The other approach computes the irreducible decomposition using the Hamburger-Noether expansion and computes the Euler characteristic in terms of the irreducible components.

Examples show that the second approach is much faster (see section-3).

Let ( ,0) C be the germ of plane curve defined by the equation 0, f where f is an analytic function of two complex variables. Then the following definitions and results can be found in [4,5,6].

Definition 1.1. The curve C defines as a line

Equations

2. THE ALGORITHMS

In this section we present the algorithms to compute the Euler characteristic of the Milnor fibre of plane curve singularity f which can be implemented in SINGULAR [2,3]

Algorithm-1: ( Eulerchrac)

Input: A polynomial , f (defines a reduced curve singularity).

Output: An integer E, the Euler characteristic of plane curve singularity.

Factorize 1 . r f f f Compute a list , the Hamburger-Noether expansion for all . i f Compute the matrix of intersection multiplicities corresponding to the irreducible branches of . f compute the Milnor number of each branch of compute the Euler characteristic of the Milnor fibre of plane curve singularity as described in Proposition 1.3. Return(E)

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Algorithm-2: ( resEuler)

Input: A polynomial , f (defines a reduced curve singularity).

Output: An integer E, the Euler characteristic of plane curve singularity.

Compute resolution graph of reduced plane curve singularity . Compute sequence of multiplicities of reduced plane curve singularity . Compute the Euler characteristic of the Milnor fibre of plane curve singular as described in Proposition 1.4. Return(E)

3. TABLE AND TIMINGS

In this section we provide some examples and a table which gives a time comparison between the 1st approach and the 2nd approach. Timings are conducted by using Singular 3-1- 3 on an Intel(R)T2400, dual core 1.83 GHz processor, 1 GB RAM under the Window 2007 operating system. We consider the following polynomials:

Equations

4. THEORETICAL COMPERISON

Now we compare the two formulae. In the 1st approach initially we make the factorization of 1 . r f f f . After this we compute the Hamburger-Noether expansion of each , i f then compute the intersection multiplicity of the branches appearing in the Hamburger-Noether expansion and also the Milnor number of each branch. In the second approach we compute the resolution graph and multiplicity sequences of f. For details see [1].

REFERENCES

[1] Binyamin,M.A.: Improving the computation of invariants of the plane curve singularies, Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matematica, vol.21, No.1 (2013), pp.51-58.

[2] Decker, W.: Greuel, G.-M.; Pfister, G.; Sch nemann, H.: Singular 3-1-1 | A computer algebra system for polynomial computations. http://www.singular.uni- kl.de (2010).

[3] Greuel, G.-M.; Pfister, G.: A Singular Introduction to Commutative Algebra. Second edition, Springer (2007).

[4] De Jong, T.; Pfister, G.: Local Analytic Geometry. Vieweg (2000).

[5] Melle-Hernandez, A.: Euler characteristic of the Milnor ibre of plane curve singularies, proceedings of the American Mathematical Society, vol.127, No.,(sep.,1999), pp.2653-2655.

[6] Nemethi,A.: Some topological invariants of isolated hypersurface singularities, EMS summer schoolEger (Hungary), 29 July-9 August (1996).
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Publication:Science International
Article Type:Report
Date:Aug 31, 2015
Words:614
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