On the computation of graded components of Laurent polynomial rings

In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let $S$ be the Laurent polynomial ring $k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_r,x_{r+1}^{\pm 1},\mathrm{.}\mathrm{.}\mathrm{.},x_n^{\pm 1}]$, $k$ algebraicaly closed field of characteristic 0. We define the multigrading of $S$ by an arbitrary finitely generated abelian group $A$. We construct a set of fans compatible with the multigrading and use this fans to compute the graded components of $S$ using polytopes. We give an algorithm to check whether the graded components of $S$ are finite dimensional. Regardless of the dimension, we determine a finite set of generators of each graded component as a module over the component of homogeneous polynomials of degree 0.