On Poincaré series associated with links of normal surface singularities

We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seiberg--Witten invariant given by the Poincaré series. We also reprove and discuss surgery formulas of Némethi for the counting function, and of Braun and Némethi for the Seiberg--Witten invariant.

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