论文标题
Hilbert功能和纯$ o $ sequisences的log-concavity
Log-concavity of level Hilbert functions and pure $O$-sequences
论文作者
论文摘要
我们在级别的Hilbert函数和纯$ o $序列的背景下调查了对数洞穴，这是Stanley在七十年代后期引入的两类数值序列，其结构性能是对组合交换代数的重要兴趣的对象。但是，由于Iarrobino的论文，对这些序列的对数洞穴的系统研究才始于最近。 本说明的目的是解决Iarrobino的工作打开的两个一般问题：1）鉴于整数$（r，t）$，所有级别的Hilbert函数是Codimension $ r $和type $ t $ log-concave的函数吗？ 2）使用相同参数的纯$ o $序列如何？ iarrobino的主要结果包括对1）的正面答案，以$ r = 2 $和任何$ t $，以及$（r，t）=（3,1）$。此外，他证明了1）的答案是$（r，t）=（4,1）$的负数。 我们对1）的主要贡献是在所有剩余案例中提供负面答案，除了$（r，t）=（3,2）$，在任何特征中仍然开放。然后，我们提出了一些专门针对Codimension 3和类型2的Hilbert功能的详细猜想。 至于问题2），我们表明答案对所有对$（r，1）$均为正面； $（r，t）=（3,4）$为负；对于任何一对$（r，t）$，带有$ r \ ge 4 $和$ 2 \ le t \ le l r+1 $的负数。有趣的是，保持打开的主要情况是$（r，t）=（3,2）$。此外，我们猜想，与任意级别的Hilbert功能的行为相比，对纯$ o $ $ $ - 任何condimension $ r \ ge 3 $ and type $ t $的行为失败。
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.
