Linear Spaces of Symmetric Matrices with Non-Maximal Maximum Likelihood Degree

$1$-generic 矩阵的极大子式具有有理奇点

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。

拉格朗日提升、交错群与最大似然度次数

Lawrence Lifts, Matroids, and Maximum Likelihood Degrees

本文通过交错群的Mobius不变量，将由拉格朗日提升完全单模矩阵参数化的托里奇品种的最大似然度次数进行了表达，并给出了无三向交互作用模型的闭式公式。

极大似然、置换多面体与结合方程

Maximum Likelihood, permutohedra and Associativity Equations

本文证明线性浓度模型对应的浓度矩阵锥是Monge–Ampère域，其对角矩阵模空间满足结合方程且紧化为置换多面体相关的环面簇，并引入Frobenius剩余来指标化极大似然度。

Reciprocal maximum likelihood degrees of diagonal linear concentration models

We show that the reciprocal maximal likelihood degree (rmld) of a diagonal linear concentration model $\mathcal L \subseteq \mathbb{C}^n$ of dimension $r$ is equal to $(-2)^rχ_M( \textstyle\frac{1}{2})$, where $χ_M$ is the characteristic polynomial of the matroid $M$ associated to $\mathcal L$. In particular, this establishes the polynomiality of the rmld for general diagonal linear concentration models, positively answering a question of Sturmfels, Timme, and Zwiernik.

矩阵乘积为零的主要原因

The main reasons for matrices multiplying to zero

本文研究给定尺寸矩阵序列乘积为零的参数簇，显式刻画其最大维分支。

Extremal behaviour in sectional matrices

In this paper we want to revive the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth and Persistence Theorem, a gen- eralization of Gotzmann's persistence Theorem. This suggests that further investigation of this object might cast a new light in the study of geometric consequences of maximal growth of the Hilbert function.

A note on spaces of symmetric matrices

We calculate the maximal dimension of linear spaces of symmetric and hermitian matrices with given high rank, generalizing a well-known result of Adams {\em et al.}

Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of $n\times n$ Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is $2n-3$.