论文标题
$ k $组件的树木生成功能的Hessian矩阵的特征值
The eigenvalues of the Hessian matrices of the generating functions for trees with $k$ components
论文作者
论文摘要
让我们考虑一个图形$γ$的图形矩阵的等级$ r $的截短的矩阵$m_γ{r} $。 $m_γ^{r} $的基础是$ r $边缘$γ$的森林集。我们认为此基础生成功能并计算其Hessian。在本文中，我们表明，完整或完整的双分部分图的截短矩阵的基础生成函数的Hessian不会通过计算Hessian矩阵的特征值而消失。此外，我们表明，完整或完整的双分部分图的截短矩阵的基本生成函数的Hessian矩阵完全具有一个正征值。作为应用程序，我们显示了与截短的Matroid相关的Artinian Gorenstein代数的强Lefschetz特性。
Let us consider a truncated matroid $M_Γ^{r}$ of rank $r$ of a graphic matroid of a graph $Γ$. The basis for $M_Γ^{r}$ is the set of the forests with $r$ edges in $Γ$. We consider this basis generating function and compute its Hessian. In this paper, we show that the Hessian of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph does not vanish by calculating the eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph has exactly one positive eigenvalue. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the truncated matroid.
