Poincaré series of a rational surface singularity (Q1434282)

In this paper, \((S,0)\) is a germ of an isolated rational surface singularity and \(\pi:(X,D)\to (S,0)\) its minimal resolution. Here, \(X\) is a smooth surface, \(\pi\) is a proper analytic map which is an isomorphism outside of \(D=\pi^{-1} (0)\), the exceptional devisor \(D\) is the union of irreducible components \(E_i\) \((i=1,\dots,r)\) transversal to each other, each component \(E_i\) is isomorphic to the projective line \(\mathbb{C}\mathbb{P}^1\). Furthermore, we will see several definitions, key-words and notations. But we have no space to introduce these definitions in this review. The authors make and prove four important calculation theorems and two propositions. At the last page of this paper, the authors apply their theorems to the \(A_k\) surface singularity and the \(D_4\) surface singularity.