论文标题
小型未成年人的决定性方面理想
Determinantal Facet Ideals for Smaller Minors
论文作者
论文摘要
确定方面的理想（DFI）是由通用$ n \ times m $矩阵的最大未成年人的子集生成的，其中$ n \ leq m $由简单复合物$Δ$的方面索引。我们考虑了$ r $ -DFI的更一般的概念，该概念是由$ r $ - 毫米的子集生成的，该子集由$δ$ of $δ$的$ 1 \ leq r \ leq r \ leq n $索引。我们定义和研究所谓的LCM封闭式和单位间隔$ r $ -DFI，并表明，相对于\ emph {Any} comph {any} compl {any} $ rubner订单，未成年人以$Δ$的构成了较少的gröbner基础。我们还看到，LCM锁定概括了先前在文献中引入的条件，并猜想在$ r = n $的情况下，LCM固定是为了成为Gröbner的基础所必需的。我们还提供有关$δ$的最大集团的条件，以确保LCM关闭和单位间隔$ r $ -dfis是Cohen-Macaulay。最后，我们以某些类型的$ r $ -DFI的贝蒂数字进行了一系列的ENE，HERZOG和HIBI的猜想，并为Cohen-Macaulay单位间隔DFI提供了证明。
A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix where $n\leq m$ indexed by the facets of a simplicial complex $Δ$. We consider the more general notion of an $r$-DFI, which is generated by a subset of $r$-minors of a generic matrix indexed by the facets of $Δ$ for some $1\leq r\leq n$. We define and study so-called lcm-closed and unit interval $r$-DFIs, and show that the minors parametrized by the facets of $Δ$ form a reduced Gröbner basis with respect to \emph{any} term order for an lcm-closed $r$-DFI. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case $r=n$, lcm-closedness is necessary for being a Gröbner basis. We also give conditions on the maximal cliques of $Δ$ ensuring that lcm-closed and unit interval $r$-DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of $r$-DFIs, and provide a proof of this conjecture for Cohen-Macaulay unit interval DFIs.
