## Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in $$\mathbb{C}^3$$.(English)Zbl 0979.32016

The main result of the paper is the following. Let $$f$$ and $$g$$ be complex quasihomogeneous polynomials with an isolated critical point at the origin. If the Seifert forms of $$f$$ and $$g$$ are equivalent over the real numbers, then the spectra of $$f$$ and $$g$$ coincide. It follows that if $$f$$ and $$g$$ have the same topological type, then the weights of $$f$$ and $$g$$ are, up to order, equal. A further corollary is the equivalence of the following seven statements for a semiquasihomogeneous function germ $$f$$ and $$g$$ in three variables.
The Seifert forms of $$f$$ and $$g$$ are equivalent over the real numbers.
The Seifert forms of $$f$$ and $$g$$ are equivalent over the integers.
$$f$$ and $$g$$ have the same characteristic polynomial and the same equivariant signatures.
$$f$$ and $$g$$ have the same spectrum.
$$f$$ and $$g$$ are connected by a $$\mu$$–constant deformation.
$$f$$ and $$g$$ have the same topological type.
The links of $$f$$ and $$g$$ have isomorphic fundamental group and $$f$$ and $$g$$ have the same characteristic polynomial. If one of the seven conditions is satisfied, then $$f$$ and $$g$$ have the same multiplicity.

### MSC:

 32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants 14B05 Singularities in algebraic geometry 32S25 Complex surface and hypersurface singularities 32S55 Milnor fibration; relations with knot theory

### Keywords:

singularities; spectrum; Seifert form
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