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Computing \(\mu^{*}\)-sequences of hypersurface isolated singularities via parametric local cohomology systems. (English) Zbl 1393.13035
Let \(X \subset \mathbb{C}^n\) be an open neighborhood of the origin \(O\) and \(f\) a holomorphic function defined on \(X\) with isolated singularity at \(O\). The Milnor number \(\mu(f)\) is the length of \(\mathcal O_{X,O}/J(f)\) where \(J(f)\) is the Jacobi ideal. For \(i = 0\), …, \(n\), let \[ \mu^{(i)}(f) = \min_L \mu(f|_L) \] where \(L\) runs over all the \(i\)-dimensional subspaces of \(\mathbb{C}^n\). The \(\mu^*\)-sequence is defined to be \[ \mu^*(f) = (\mu^{(n)}(f), \dots, \mu^{(0)}(f)). \] In the present paper, authors give a new algorithm to compute \(\mu^*(f)\). It is based on author’s preceding paper [in: Proceedings of the 39th international symposium on symbolic and algebraic computation, ISSAC 2014, Kobe, Japan, July 23–25, 2014. New York, NY: Association for Computing Machinery (ACM). 351–358 (2014; Zbl 1325.68297)]. Furthermore this paper contains a table of \(\mu^*\)-sequence of famous singularties.

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
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References:
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