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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-1)$$\end{document}th and (n-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-2)$$\end{document}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.