Hilbert polynomials and module generating degrees

TL;DR

This paper generalizes classical theorems relating Hilbert polynomials and module properties by incorporating generating degrees, leading to sharper bounds and broader applicability in algebraic geometry.

Contribution

It introduces a new form of Gotzmann representation based on rank and generating degrees, extending regularity bounds and generalizing Macaulay and Green's theorems.

Findings

Established a generalized Gotzmann representation for modules.

Proved sharpness of the Gotzmann regularity bound under certain conditions.

Extended Macaulay and Green's theorems through analogous modifications.

Abstract

We establish a form of the Gotzmann representation of the Hilbert polynomial based on rank and generating degrees of a module, which allow for a generalization of Gotzmann's Regularity Theorem. Under an additional assumption on the generating degrees, the Gotzmann regularity bound becomes sharp. An analoguous modification of the Macaulay representation is used along the way, which generalizes the theorems of Macaulay and Green, and Gotzmann's Persistence Theorem.