How to compute the multigraded Hilbert depth of a module

The aim of this paper is to introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module $M$ over the polynomial ring $K[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n]$ by reducing the problem to the computation of the finite set of the new defined Hilbert partitions. Moreover, in the last section, we show that Hilbert partitions may also be used for computing the Stanley depth of the module $M$.