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
Restricted to Repository staff only until 31 December 2020.
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.
|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|
K. Altmann, Equisingular deformations of isolated 2-dimensional hypersurface singularities, Invent. Math. 88 (1987) 619-634.
V. I. Arnol'd, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps - Vol.II (Birkhäuser, Boston, MA, 1988).
E. Artal Bartolo, Forme de Seifert des singularités de surface,C. R.Acad.Sci.Paris Sér. I Math. 313 (1991) 689-692.
E. Artal Bartolo, P. Cassou-Noguès, I. Luengo and A. Melle-Hernández, Monodromy conjecture for some surfaces, Ann. Sci. École Norm. Sup. (4), to appear.
J. Denef and F. Loeser, Caractéristiques d'Euler-Poincaré, fonctions zeta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992) 705-720.
J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998) 505-537.
J. Denef and F. Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (1999) 285-309.
P. Du Bois and F. Michel, The integral Seifert form does not determine the topology of plane curve germs, J. Algebraic Geom. 3 (1994) 1-38.
A. H. Durfee, Fibered knots and algebraic singularities, Topology 13 (1974) 47-59.
F. Loeser, Fonctions d'Igusa p-adiques et polynomes de Bernstein, Amer. J. Math. 110 (1988) 1-21.
F. Loeser, Fonctions d'Igusa p-adiques, polynomes de Bernstein, et polyedres de Newton, J. Reine Angew. Math. 412 (1990) 75-96.
B. Rodrigues and W. Veys, Holomorphy of Igusa's and topological zeta functions for homogeneous polynomials', Pacic J. Math., to appear.
K. Sakamoto, The Seifert matrices of Milnor berings dened by holomorphic functions, J. Math. Soc. Japan 26 (1974) 714-721.
W. Veys, Determination of the poles of the topological zeta function for curves, Manuscripta Math.87 (1995) 435-448.
W. Veys, Zeta functions for curves and log canonical models, Proc. London Math. Soc. (3) 74 (1997) 360-378.
T. Yano, On the theory of b-functions, Publ. Res. Inst. Math. Sci. 14 (1978) 111-202.
|Deposited On:||16 Oct 2012 10:20|
|Last Modified:||16 Oct 2012 10:20|
Repository Staff Only: item control page