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.