Abstract
We investigate log-concavity in the context of level Hilbert functions and pure -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 , are all level Hilbert functions of codimension and type log-concave? (2) What about pure -sequences with the same parameters?
Iarrobino’s main results consisted of a positive answer to (1) for and any , and for . Further, he proved that the answer to (1) is negative for .
Our chief contribution to (1) is to provide a negative answer in all remaining cases, with the exception of , 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 , negative for , and negative for any pair with and . Interestingly, the main case that remains open is again . Further, we conjecture that, in analogy with the behavior of arbitrary level Hilbert functions, log-concavity fails for pure -sequences of any codimension and type large enough.
Citation
Fabrizio Zanello. "LOG-CONCAVITY OF LEVEL HILBERT FUNCTIONS AND PURE -SEQUENCES." J. Commut. Algebra 16 (2) 245 - 256, Summer 2024. https://doi.org/10.1216/jca.2024.16.245
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