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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
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