A polynomial counterpart of the Seiberg-Witten invariant associated with a negative definite
plumbing 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.