On the sign patterns of the coefficients of Hilbert polynomials

TL;DR

This paper proves that any sign pattern of coefficients in Hilbert polynomials of standard graded algebras can occur, demonstrating the diversity of coefficient signs.

Contribution

It establishes the universality of sign patterns in Hilbert polynomial coefficients for standard graded algebras.

Findings

All sign patterns are realizable in Hilbert polynomials.

The result applies to any sign combination of coefficients.

It broadens understanding of Hilbert polynomial coefficient behavior.

Abstract

We show that all sign patterns of the coefficients Hilbert polynomials of standard graded $k$-algebras are possible.