LOG-CONCAVITY OF LEVEL HILBERT FUNCTIONS AND PURE O-SEQUENCES

Abstract

We investigate log-concavity in the context of level Hilbert functions and pure $O$-sequences, two classes of numerical sequences introduced by Stanley in the late 1970s whose structural properties have since been the object of a remarkable amount of interest in combinatorial commutative algebra. However, a systematic study of the log-concavity of these sequences began only recently, thanks to a paper by Iarrobino.

The goal of this note is to address two general questions left open by Iarrobino’s work: (1) Given the integer pair $(r,t)$, are all level Hilbert functions of codimension $r$ and type $t$ log-concave? (2) What about pure $O$-sequences with the same parameters?

Iarrobino’s main results consisted of a positive answer to (1) for $r=2$ and any $t$, and for $(r,t)=(3,1)$. Further, he proved that the answer to (1) is negative for $(r,t)=(4,1)$.

Our chief contribution to (1) is to provide a negative answer in all remaining cases, with the exception of $(r,t)=(3,2)$, which is still open in any characteristic. We then propose a few detailed conjectures specifically on level Hilbert functions of codimension 3 and type 2.

As for question (2), we show that the answer is positive for all pairs $(r,1)$, negative for $(r,t)=(3,4)$, and negative for any pair $(r,t)$ with $r\ge 4$ and $2\le t\le r+1$. Interestingly, the main case that remains open is again $(r,t)=(3,2)$. Further, we conjecture that, in analogy with the behavior of arbitrary level Hilbert functions, log-concavity fails for pure $O$-sequences of any codimension $r\ge 3$ and type $t$ large enough.