In this dissertation, we will discuss some properties of pure O-sequences, which, due to Macaulay’s Inverse Systems, are in bijective correspondence with the Hilbert functions of Artinian level monomial algebras. In particular, we will focus on their unimodality. A sequence of numbers is unimodal if it does not increase after a strict decrease.
There has been previous progress in this area. Specifically, it is known that all algebras in two variables are unimodal, due to Macaulay’s Maximal Growth Theorem. Furthermore, it is known that all pure O-sequences of type two in three variables are unimodal and the Hilbert function of monomial complete intersections are unimodal; these results are due to the Weak Lefschetz Property. In addition to these families of pure O-sequences which are unimodal, there are known families of pure O-sequences which fail to be unimodal. In particular, for any r > 2, there exists a monomial Artinian level algebra in r variables whose Hilbert function fails unimodality with an arbitrary number of peaks.
We will focus on the question of whether particular socle types or socle degrees in a fixed number of variables guarantee unimodality. Since the Weak Lefschetz Property is only guaranteed to hold in the cases mentioned above, we will use different techniques in our approach. We will show that all pure O-sequences of type three in three variables and all pure O-sequence of type two in four variables are strictly unimodal. We will also show that all pure O-sequences with socle degree less than or equal to nine in three variables are unimodal and all pure O-sequences (except for possibly two cases) with socle degree less than or equal to four in four variables are unimodal. Finally we will show that for r greater than or equal to 4 and e greater than or equal to 7, there exists a non-unimodal pure O-sequence in r variables with socle degree e.