On minimal Gorenstein Hilbert function

TL;DR

This paper investigates a conjecture that certain Artinian Gorenstein algebras, called full Perazzo algebras, always have minimal Hilbert functions, providing proofs for specific lengths and codimensions and offering new insights into their asymptotic behavior.

Contribution

It proves the conjecture for lengths four and five in low codimension and for a specific subclass across all lengths and some codimensions, advancing understanding of Gorenstein Hilbert functions.

Findings

Confirmed the conjecture for length four and five in low codimension.

Proved the conjecture for a subclass of algebras in all lengths and certain codimensions.

Provided a new proof regarding the asymptotic behavior of the minimum Gorenstein Hilbert function entry.

Abstract

We conjecture that a class of Artinian Gorenstein Hilbert algebras called full Perazzo algebras always have minimal Hilbert function, fixing codimension and length. We prove the conjecture in length four and five, in low codimension. We also prove the conjecture for a particular subclass of algebras that occurs in every length and certain codimensions. As a consequence of our methods we give a new proof of part of a known result about the asymptotic behavior of the minimum entry of a Gorenstein Hilbert function.