Log-concavity of level Hilbert functions and pure $O$-sequences
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| Formato: | Preprint |
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2022
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| _version_ | 1866911878879379456 |
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| author | Zanello, Fabrizio |
| author_facet | Zanello, Fabrizio |
| contents | 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 Seventies 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) How 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_09447 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Log-concavity of level Hilbert functions and pure $O$-sequences Zanello, Fabrizio Commutative Algebra Combinatorics Primary: 13D40, Secondary: 05E40, 13E10, 13H10 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 Seventies 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) How 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. |
| title | Log-concavity of level Hilbert functions and pure $O$-sequences |
| topic | Commutative Algebra Combinatorics Primary: 13D40, Secondary: 05E40, 13E10, 13H10 |
| url | https://arxiv.org/abs/2210.09447 |