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On `maximal' poles of zeta functions, roots of b-functions and monodromy jordan blocks

Melle Hernández, Alejandro and Torrelli , Tristan and Veys, Willen (2009) On `maximal' poles of zeta functions, roots of b-functions and monodromy jordan blocks. Journal of Topology , 2 (3). pp. 517-526. ISSN 1753-8416

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Official URL: http://jtopol.oxfordjournals.org/

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Abstract

The main objects of this study are the poles of several local zeta functions: the Igusa, topological, and motivic zeta function associated to a polynomial or (germ of) holomorphic function in n variables. We are interested in poles of maximal possible order n. In all known cases (curves, non-degenerate polynomials) there is at most one pole of maximal order n, which is then given by the log canonical threshold of the function at the corresponding singular point. For an isolated singular point we prove that if the log canonical threshold yields a pole of order n of the corresponding (local) zeta function, then it induces a root of the Bernstein-Sato polynomial of the given function of multiplicity n, (proving one of the cases of the strongest form of a conjecture of Igusa-Denef-Loeser). For an arbitrary singular point, we show under the same assumption that the monodromy eigenvalue induced by the pole has 'a Jordan block of size n on the (perverse) complex of nearby cycles'.

Item Type:Article
Uncontrolled Keywords:Topological zeta functions; Motivic zeta functions; Monodromy; Bernstein-Sato polynomial
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:13440
Deposited On:19 Oct 2011 07:24
Last Modified:06 Feb 2014 09:49

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