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Zeta-functions for germs of meromorphic functions and Newton diagrams

Melle Hernández, Alejandro and Gusein-Zade, Sabir Medgidovich and Luengo Velasco, Ignacio (1998) Zeta-functions for germs of meromorphic functions and Newton diagrams. Functional Analysis and its applications , 32 (2). 93 -99. ISSN 0016-2663

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Abstract

Let f be a meromorphic function germ on (Cn+1, 0); that is, f = P/Q, where P,Q: (Cn+1, 0)! (C, 0) are holomorphic germs. The authors introduce a notion of Milnor fibers and monodromy operators of the germ f around zero and infinity. Based on their previous work [Comment. Math. Helv. 72 (1997), no. 2, 244–256; MR1470090 (98j:32043)] they write down formulas for the zetafunctions of the monodromy operators in terms of partial resolutions of a singularity. In the case where P and Q are non-degenerate relative to their Newton’s diagrams an analog of the formula from [A. N. Varchenko, Invent. Math. 37 (1976), no. 3, 253–262; MR0424806 (54 #12764)] for zeta functions of monodromy operators is obtained. In conclusion, two interesting examples with f = (x3 −xy)/y and f = (xyz +xp +yq +zr)/(xd +yd +zd) are discussed in detail.


Item Type:Article
Uncontrolled Keywords:Monodromy; Germ of meromorphic function; Resolution of germ; Milnor fiber; Resolution of singularities; Monodromy transformation; Zeta function of monodromy; A'Campo formula; partial resolution; Newton diagrams
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:12904
Deposited On:30 Jun 2011 07:38
Last Modified:06 Feb 2014 09:35

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