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

We prove that the Hilbert functions of Gorenstein Artin algebras R/I of embedding dimension four are unimodal provided I has a minimal generator in degree less than five. It is still an open question as to whether all Gorenstein Hilbert functions in codimension four are SI-sequences. It is not even known if they are all unimodal. In this article, we prove that Hilbert functions of all Gorenstein Artin algebras starting with (1, 4, 10, 20, h 4,…), where h 4 = 34 are unimodal. Combining this with previously known results, we obtain that all Gorenstein Hilbert functions (1, 4, h 2, h 3, h 4,…4, 1) are unimodal if h 4 ≤ 34.