The intrinsic topological nature of the Poincaré series of a plane curve singularity

In this paper we provide some factorization theorems of the Poincaré series $P_C$ of a plane curve singularity $C$ depending on some key values of the semigroup of values of $C$. These results yield an iterative computation of $P_C$ in purely algebraic terms from the dual resolution graph of $C$. On the other hand, Campillo, Delgado and Gusein-Zade showed in 2003 the equality between $P_C$ and the Alexander polynomial ${\mathrm{\Delta }}_L$ of the corresponding link $L$. Our procedure supplies a new proof of this coincidence. More concretely, we show that our algebraic construction can be translated to the iterated toric structure of the link $L$. Additionally we show that the semigroup algebra can be defined from the fundamental group of the link exterior in the irreducible case. This gives in particular a conceptual reason for the coincidence of $P_C$ and ${\mathrm{\Delta }}_L$.