Arnold's seminar
April 9, 2002

S.Gusein-Zade
Zeta-function and Poincare series of a hypersurface singularity.

A.Campillo, F.Delgado and S.Gusein-Zade have noticed that
the Poincare series of an irreducible plane curve singularity
(with respect to a natural filtration on it) coincided with
the zeta-function of its classical monodromy transformation.
There were generalisations of this fact to (global) algebraic
plane curves with one place at infinity and to reducible plane
curve singularities. All these facts are proved by direct
computations of both sides of an equation. There is no explanation
for any of them. Later W.Ebeling described a connection
between the Poincare series of an isolated quasihomogeneous
2-dimensional hypersurface singularity (with respect to the
quasihomogeneous filtration) and the characteristic polynomial
of the monodromy transformation (the relation includes Saito
duality between cyclotomic polynomials and orbit invariants
of the $C^*$ action on the singularity; in some sense one can
say that in the intersection of both statements there are
irrreducible quasihomogeneous plane curve singularities
$x^a+y^b=0$). We describe a generalisation of the last statement
to Newton non degenerate quasihomogeneous hypersurface singularities
of any dimension. This clarifies the initial result of W.Ebeling.
This is the result of a joint work of W.Ebeling and S.Gusein-Zade.