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The Denef-Loeser zeta function is not a topological invariant

Artal Bartolo, Enrique and Cassou-Noguès, Pierrette and Luengo Velasco, Ignacio and Melle Hernández, Alejandro (2002) The Denef-Loeser zeta function is not a topological invariant. Journal of the London Mathematical Society. Second Series, 65 (1). pp. 45-54. ISSN 0024-6107

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Abstract

An example is given which shows that the Denef–Loeser zeta function (usually called the topological zeta function) associated to a germ of a complex hypersurface singularity is not a topological invariant of the singularity. The idea is the following. Consider two germs of plane curves singularities with the same integral Seifert form but with different topological type and which have different topological zeta functions. Make a double suspension of these singularities (consider them in a 4-dimensional complex space). A theorem of M. Kervaire and J. Levine states that the topological type of these new hypersurface singularities is characterized by their integral Seifert form. Moreover the Seifert form of a suspension is equal (up to sign) to the original Seifert form. Hence these new singularities have the same topological type. By means of a double suspension formula the Denef–Loeser zeta functions are computed for the two 3-dimensional singularities and it is verified that they are not equal.


Item Type:Article
Additional Information:

During the development of this paper, the second author was
the guest of the Department of Algebra at the University Complutense of Madrid, supported by a sabbatical grant from the MEC. She wishes to thanks the MEC for its support and the members of the Department of Algebra for their warm
hospitality.

Uncontrolled Keywords:Adic Igusa Functions; Bernstein Polynomials; Curves
Subjects:Sciences > Mathematics > Functions
ID Code:16722
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Deposited On:16 Oct 2012 08:20
Last Modified:07 Feb 2014 09:34

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