Weight filtrations on GKZ-systems

Given an integer matrix $A\in\Bbb{Z}^{d\times n}$, we study the natural mixed Hodge module structure in sense of Saito on Gau\ss--Manin system attached to monomial map $h\colon(\Bbb{C}^*)^d\to\Bbb{C}^n$ induced by $A$. We completely determine normal case associated graded object weight filtration, computing intersection complexes with respective multiplicities that form its constituents. Our results show these data are purely combinatorial, and not arithmetic, they only depend polyhedral cone $A$, but semigroup itself. In particular, extend de Cataldo, Migliorini Musta\c{t}\v{a} setting torus embeddings give a closed for failure Decomposition Theorem our context.