N\'emethi's division algorithm for zeta-functions of plumbed 3-manifolds

A polynomial counterpart of the Seiberg-Witten invariant associated with a negative definite plumbed 3-manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta-function defined by the combinatorics of the manifold. In this article we give an algorithm, based on multivariable Euclidean division of the zeta-function, for the explicit calculation of the polynomial, in particular for the Seiberg--Witten invariant.