Permanental ideals of Hankel matrices
Elena Grieco, Anna Guerrieri, Irena Swanson
给出了Hankel矩阵的2 × 2积和式生成的理想的Grobner基和准素分解.
We provide the Grobner basis and the primary decomposition of the ideals generated by 2 by 2 permanents of Hankel matrices.
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The v-numbers of permanental ideals
本文计算了通用、通用对称和通用Hankel矩阵的2×2 permanental ideals的v数。
Minimal Primes Over Permanental Ideals
本文讨论一般矩阵的永久理想上的极小素性。给出了一般矩阵的3 × 3积和式理想上的最小素数的完整列表，并证明了当基域的特征充分大时，d × 2d-1矩阵的最大积和式理想中存在单项式.我们还讨论了Alon-Jaeger-Tarsi猜想，利用我们的结果和技巧加强了以前已知的结果。
Permanental Ideals
主要结果是一般矩阵的（2 × 2）-子积式所生成的理想的准素分解。这些永久理想几乎总是有嵌入的成分，它们的极小素数有三个不同的高度。因此，与行列式理想相比，恒常理想几乎从来不是科恩-麦考利理想。
2 x 2 permanental ideals of hypermatrices
研究了超矩阵的几类2 \times 2积和式生成的理想的结构。这推广了[9]关于一般矩阵的2 × 2持久理想的结果.我们将得到的结构与[Swanson-Taylor 11]中相应的行列式理想的结构进行比较：虽然[11]中引入的t-可切换性的概念对持久理想和行列式理想都起作用，但持久理想需要进一步的限制，这通常会增加最小素数的数量。在最后两节中，我们考察了几个相关的永久理想类。这是第一作者2011年在里德学院的高级论文的延伸，在第二作者的监督下。
We give an explicit formula for the Hilbert-Poincaré series of the parity binomial edge ideal of a complete graph $K_{n}$ or equivalently for the ideal generated by all $2\times 2$-permanents of a $2\times n$-matrix. It follows that the depth and Castelnuovo-Mumford regularity of these ideals are independent of $n$.
Permanental ideals of symmetric matrices
本文研究对称矩阵的 $2\ imes 2$ 永久子式生成的理想 $P_2(X)$，计算其 Gröbner 基、维数、深度、极小素理想与准素分解，并揭示其代数性质依赖于基域特征是否为二。
In this paper, we study the Hankel edge ideals of graphs. We determine the minimal prime ideals of the Hankel edge ideal of labeled Hamiltonian and semi-Hamiltonian graphs, and we investigate radicality, being a complete intersection, almost complete intersection and set theoretic complete intersection for such graphs. We also consider the Hankel edge ideal of trees with a natural labeling, called rooted labeling. We characterize such trees whose Hankel edge ideal is a complete intersection, and moreover, we determine those whose initial ideal with respect to the reverse lexicographic order sa…
The determinantal ideals of extended Hankel matrices
本文利用Gröbner基和组合割线簇的工具研究了广义Hankel矩阵的行列式理想$I_t$.用$c$ -链表示序列$a_1,\...,a_k$，其中对于所有$i=1,\...,k-1$具有$a_i+c<a_{i+1}$。利用$c$ -链的结果，解决了符号幂$I_t^{(s)}$的隶属度问题，并计算了行列式理想的乘积$I_{t_1}\... I_{t_k}$的准素分解.通过初始理想和代数，证明了乘积$I_{t_1}\... I_{t_k}$具有线性分解，多齐次Rees代数$\Rees(I_{t_1},\...,I_{t_k})$由二次曲面的Gröbner基定义.
We introduce the concept of s-Hankel hypermatrix, which generalizes both Hankel matrices and generic hypermatrices. We study two determinantal ideals associated to an s-Hankel hypermatrix: the ideal I<s,t> generated by certain 2 x 2 slice minors, and the ideal \tilde{I}<s,t> generated by certain 2 x 2 generalized minors. We describe the structure of these two ideals, with particular attention to the primary decomposition of I<s,t>, and provide the explicit list of minimal primes for large values of s. Finally we give some geometrical interpretations and generalise a theorem of Watanabe.
Ideals generated by $2$--minors of Hankel matrices
我们研究理想产生的$2$ -未成年人的一般汉克尔矩阵。
Motivated by a work of Knutson, in a recent paper Conca and Varbaro have defined a new class of ideals, namely "Knutson ideals", starting from a polynomial $f$ with squarefree leading term. We will show that the main properties that this class has in polynomial rings over fields of characteristic $p$ are preserved when one introduces the definition of Knutson ideal also in polynomial rings over fields of characteristic zero. Then we will show that determinantal ideals of Hankel matrices are Knutson ideals for a suitable choice of the polynomial $f$.
In this paper, we study the family of determinantal ideals of "close" cuts of Hankel matrices, say $ \mathcal{f} $. We show that the multi-Rees algebra of ideals in $ \mathcal{f} $ is defined by a quadratic Gröbner basis, it is Koszul, normal Cohen-Macaulay domain and it has a nice Sagbi basis. As a consequence of Koszulness, we prove that every product $ I_1\ldots I_l $ of ideals of $ \mathcal{f} $ has linear resolution. Moreover, we show that natural generators of every product $ I_1\ldots I_l $ form a Gröbner basis.
Dimensions of permanental varieties in arbitrary size
本文证明任意尺寸下永久型簇的维数公式，并得出永久式临界轨迹的余维数及极大永久式生成完全交的约化性。
Special fibers of coordinate sections of Hankel Matrices
本文研究Hankel矩阵坐标截面的特殊纤维，揭示其定义方程的结构特征，并证明其Cohen-Macaulay性质及极小生成元的次数增长规律。
One deals with degenerations by coordinate sections of the square generic Hankel matrix over a field $k$ of characteristic zero, along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these derivatives, the Hessian matrix and the ideal of the submaximal minors of the matrix. It is proved that the polar map is dominant for any such degenerations, and not homaloidal in the generic case. The problem of whether the determinant $f$ of the matrix is a factor of the Hessian with the (Segre) expected multi…
Gorenstein binomial edge ideals associated with scrolls
设$I_G$是与顶点集[n]上的闭图$G$相关联的一般2 × n - Hankel矩阵上的二项式边理想.我们刻画了图$G$，其中$I_G$具有最大正则性，并且是Gorenstein图.
Binomial edge ideals and rational normal scrolls
设$X$为$2\times n$的Hankel矩阵，$G$为$[n].$上的闭图，研究了$X$的所有$2$ -子式生成的二项式理想$I_G\subset K[x_1,\ldots,x_{n+1}]$，这些子式对应于$G.$的边，证明了$I_G$是Cohen-Macaulay理想.我们找到了$I_G$的极小素数，并证明了$I_G$是一个集合论完全交.此外，还给出了$I_G$正则性的一个精确上界.
Let $R=\mathbb{K}[x_1,\dots,x_n]$ and let $\mathfrak{a}_1,\dots,\mathfrak{a}_m$ be homogeneous ideals satisfying certain properties, which include a description of the Noetherian symbolic Rees algebra. We give a solution to a question of Harbourne and Huneke for this set of ideals. We also compute the Waldschmidt constant and resurgence and show that it exhibits a stronger version of the Chudnovsky and Demailly-type bounds. We further show that these properties are satisfied for classical varieties such as the generic determinantal ideals, minors of generic symmetric matrices, generic extended…
We study the symbolic powers of determinantal ideals of generic, generic symmetric, and Hankel matrices of variables, and of Pfaffians of generic skew-symmetric matrices, in prime characteristic. Specifically, we show that the limit $\lim\limits_{n\to\infty} \frac{\textrm{reg}(I^{(n)})}{n}$ exists and that $\textrm{depth}(R/I^{(n)})$ stabilizes for $n\gg 0$. Furthermore, we give explicit formulas for the stable value of $\textrm{depth}(R/I^{(n)})$ in the generic and skew-symmetric cases. In order to show these results, we introduce the notion of symbolic $F$-purity of ideals which is satisfied…
The $F$-singularities of algebras defined by permanents
本文研究由永久式理想定义的代数的F-奇点，证明了在特征大于2时n×n永久式理想代数为F-正则，并完整刻画了2×2情形下F-纯与F-正则的条件，结果与特征2不同。
Maximal minors of $1$-generic matrices have rational singularities
本文证明 $1$-generic 矩阵极大子式理想的商环具有有理奇点，证实 Eisenbud 的正规性猜想并推广 generic Hankel 行列式环的相关结论。
Algebras of minors
设$X$是域$K$（具有足够大的特征）上的待定矩阵的$n\times m$，$M_t$是$X$的$m$-子式的集合.我们考虑两个对象：（1）多项式环$K[X]$关于由$M_t$生成的理想$I_t$的Ress代数，（2）由$M_t$生成的$K[X]$的$A_t$子代数.注意，$A_t$是格拉斯曼环的坐标环，如果$t=\min(m,n)$;同样，$t=1$和$t=m-1=n-1$的情况也很容易理解，因为在这些情况下，$A_t$是$K$上的多项式环。对于这两个对象，我们计算除数类组和规范类。特别是我们确定的Gorenstein环之间的$A_t$。事实证明，$A_t$是Gorenstein完全在上述情况下，当$t(m+n)=mn$。我们使用初始化方法，基于矫直定律和KRS。它们也可以应用于其他类型的行列式理想。我们这样做显式通用汉克尔矩阵。
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