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Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras. (English) Zbl 1305.32016
For any complex analytic isolated hypersurface singularity in $$({\mathbb C}^n,0)$$ one can consider its moduli (or Tjurina) algebra. By a theorem due to J. N. Mather and S. S. T. Yau [Invent. Math. 69, 243–251 (1982; Zbl 0499.32008)], for any fixed $$n$$, two isolated hypersurface singularities are biholomorphically equivalent if and only if their moduli algebras are isomorphic. The proof of the Mather-Yau theorem does not provide any explicit procedure for the reconstruction of the germs from their moduli algebras. The present article provides such an explicit method for the construction of the singular germ whenever the singularity is homogeneous. In this case the moduli algebra equals the Milnor algebra, and the dimension $$n$$ also can be recovered from the algebra as its embedding dimension. The method is illustrated on a concrete example.

##### MSC:
 32S25 Complex surface and hypersurface singularities 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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