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We give a formula connecting the Poincaré series of the coordinate ring of a (Newton non-degenerate) quasihomogeneous hypersurface singularity with the Saito dual of the zeta function of its monodromy. In the last years there have appeared several results which can be described as follows: the Poincaré series (or rather a Poincaré series: sometimes one has to be careful with the choice) of the ring of functions on a singularity is connected with the zeta function of its classical monodromy transformation. The first observation of this sort [CDG1] says that the Poincaré series of an irreducible plane curve singularity (with respect to the natural filtration defined by a uniformization of the curve) coincides with the zeta function of its classical monodromy transformation. There was obtained a global version of this statement for an algebraic plane curve with one place at infinity [CDG2]. For a reducible plane curve singularity, it was shown that the Poincaré polynomial (appropriately defined as a polynomial in several variables) of the multi-index filtration of the ring of functions (defined by uniformizations of the branches of the curve) coincides with the Alexander polynomial ∆(t1, ..., tr) (in several variables) of the link of the singularity [CDG3, CDG4]. (The Alexander polynomial reduces to the zeta function of the monodromy when identifying all the variables ti: ζ(t) = ∆(t, . . . , t).) It was noticed that this statement has a formulation in terms similar to motivic integration [CDG5]. For an isolated quasihomogeneous two-dimensional hypersurface singularity, it was shown that the Poincaré series of the (quasihomogeneous) filtration on the ring of functions on the singularity can be obtained from the characteristic polynomial of the classical monodromy operator by applying the Saito duality construction and multiplying by binomials corresponding to the orbit invariants of the natural action of the group C∗ of nonzero complex numbers on the surface [E1, E2]. Saito duality [S1, S2] was formulated in order to describe the connection between the characteristic polynomials of the dual singularities in Arnold’s Received March 18, 2002. 2000 Mathematics Subject Classification. 14B05, 32S40.

Poincaré series and zeta function of the monodromy of a quasihomogeneous singularity