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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 ≥ 4 and 2 ≤ t ≤ 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 ≥ 3 and type t large enough.

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Journal of Commutative Algebra