Zeta functions for germs of meromorphic functions, and Newton diagrams | Zendy

S. M. Guseĭn-Zade | Zendy; I. Luengo | Zendy; A. Melle–Hernàndez | Zendy

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Zeta functions for germs of meromorphic functions, and Newton diagrams

Author(s) -

S. M. Guseĭn-Zade,

I. Luengo,

A. Melle–Hernàndez

Publication year - 1998

Publication title -

functional analysis and its applications

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.413

H-Index - 28

eISSN - 1573-8485

pISSN - 0016-2663

DOI - 10.1007/bf02482595

Subject(s) - monodromy , meromorphic function , mathematics , holomorphic function , l function , pure mathematics , infinity , singularity , isolated singularity , riemann zeta function , analytic function , algebra over a field , combinatorics , mathematical analysis

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

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