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

TL;DR

This paper studies Hilbert quasi-polynomials in non-standard graded rings, characterizing their structure, degree, coefficients, and periodicity, and provides software for their computation.

Contribution

It provides a complete characterization of the lower degree quasi-polynomial T within Hilbert quasi-polynomials and describes its periodic structure, along with an implementation for computation.

Findings

Determined the degree of the quasi-polynomial T

Identified the first few coefficients of the Hilbert quasi-polynomial P

Described the periodic structure of T

Abstract

The Hilbert function, its generating function and the Hilbert polynomial of a graded ring R have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen [Hil90]. In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T. We have completely determined the degree of T and the first few coefficients of P. Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also implemented a software to compute effectively the Hilbert quasi-polynomial for any quotient ring R/I.