Zeta Functions

September 18 - 22, 2006, Moscow, Russia

Laboratoire J.-V. Poncelet

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Sabir Gussein-Zade

Independent University of Moscow, Russia

Monodromy zeta functions and Poincare series of filtrations on ring of functions.

The first observation which related the objects mentioned in the title was the following one. Let $(C,0)$ be an irreducible germ of a plane curve at the origin in the complex plane, given by the equation $f=0$. The zeta function of the classical monodromy transformation of the germ $f$ is a topological invariant of it. Let us consider a parametrization (uniformization) of the curve $(C,0)$. The order of a function on $(C,0)$ as a function in the uniformization parameter defines a decreasing filtration on the ring of functions on the curve singularity. It appears that the Poincare series of this filtration (considered as a rational function) coincides with the zeta function mentioned above. Now there are a number of generalizations of this statement. All of them are obtained by direct computations of the monodromy zeta functions and of the Poincare series of the filtrations. A notion of an integration with respect to the Euler characteristic over the projectivization of the ring of function is very useful for computating the Poincare series. The talk reflects joint works with A.Campillo and F.Delgado.