Determination of the poles of the topological zeta function for curves |
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Authors: | Willem Veys |
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Institution: | (1) Department Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium |
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Abstract: | Tof ∈ℂx
1…,x
n
] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f
−1{0} inf
−1{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local
zeta function. A major problem is the study of its poles.
In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂx
1,x
2]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs
rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution.
The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research
N.F.W.O. |
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Keywords: | 14B05 14H20 32S45(11S40) |
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