A partial characterization of Hilbert quasi-polynomials in the non-standard case

In this paper, we present some work towards a complete characterization of Hilbert quasi-polynomials of graded polynomial rings. In this setting, a Hilbert quasi-polynomial splits in a polynomial F and a lower degree quasi-polynomial G. We completely describe the periodic structure of G. Moreover, we give an explicit formula for the (n- 1) th and (n- 2) th coefficient of F, where n denotes the degree of F. Finally, we provide an algorithm to compute the Hilbert quasi-polynomial of any graded polynomial ring.