On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four

TL;DR

This paper proves that Hilbert functions of certain codimension four Gorenstein Artin algebras are unimodal, especially when the initial degrees are less than five or when specific h-vector conditions are met, advancing understanding of their structure.

Contribution

The authors establish unimodality of Hilbert functions for codimension four Gorenstein Artin algebras under new conditions, including minimal generators in low degrees and bounds on initial h-vector entries.

Findings

Hilbert functions are unimodal if I has a generator in degree less than five.

All Gorenstein h-vectors with h_4 ≤ 3 are unimodal.

Unimodality holds for h_4 ≤ 34 in specific initial degree sequences.

Abstract

We prove that the Hilbert functions of codimension four graded Gorenstein Artin algebras R/I are unimodal provided I has a minimal generator in degree less than five. It is an open question whether all Gorenstein h-vectors in codimension four are unimodal. In this paper, we prove that Hilbert functions of all artinian codimension four Gorenstein algebras starting with (1,4,10, 20, h_4,...), where h_4\leq 34 are unimodal. Combining this with the previously known results, we obtain that all Gorenstein h-vectors (1, h_1, h_2,h_3, h_4, ....) are unimodal if h_4\leq 3.