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Alternative algorithms for computing generic \(\mu^\ast\)-sequences and local Euler obstructions of isolated hypersurface singularities. (English) Zbl 1423.13097
Let \(f\in \mathbb{C}\{x_1, \ldots, x_n\}\). The Milnor number of \(f\) is defined by \(\mu(f)=\dim_{\mathbb{C}}\mathbb{C}\{x_1, \ldots, x_n\}/\langle\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n}\rangle\). B. Teissier [Astérisque 7–8, 285–362 (1974; Zbl 0295.14003); Invent. Math. 40, 267–292 (1977; Zbl 0446.32002)] introduced the \(\mu^\ast\)-sequence \((\mu^0(f), \ldots, \mu^{(n)}(f))\) defined by \(\mu^{(0)}(f)=1\) and \(\mu^{(i)}(f)=\min\limits_{L} \mu(f)|_L)\) where \(L\) runs over the \(i\)-dimensional linear subspaces of \(\mathbb{C}^n\) and \(f|_L\) is the restriction of \(f\) to \(L\).
A new algorithm is introduced for computing the \(\mu^\ast\)-sequence. The algorithm is implemented in the computer algebra system Singular. Timings are given proving that the algorithm has a very good performance.

MSC:
13D45 Local cohomology and commutative rings
32C37 Duality theorems for analytic spaces
13J05 Power series rings
32A27 Residues for several complex variables
Software:
Risa/Asir; SINGULAR
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References:
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