切割和分裂的复曲面几何形状

Splittings of Toric Ideals

Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in …

极小分解的典范理想

Radical splittings of toric ideals

本文研究了典范理想极小分解数的计算及其与二项式算术秩的关系，特别针对完全二分图的典范理想证明了极小分解数为 3。

Toric 分裂

Toric splittings

本文研究 toric 理想的可分裂性，利用矩阵 $A$ 给出充要判据，并据此判定多类 toric 理想是否可分裂。

齐次环面理想的粘合与分裂

Gluing And Splitting of Homogeneous Toric Ideals

本文研究齐次仿射半群的粘合，证明其环面理想之和仍为齐次环面理想且极小分次自由分解为张量积，并用于刻画图与超图环面理想的分裂。

图的环面理想的分裂

Splittings of toric ideals of graphs

本文研究简单图环面理想的子图分裂，证明其等价于边可分裂，并刻画完全二部图、完全图及极小分裂的性质。

Combinatorial degree bound for toric ideals of hypergraphs

Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform hypergraphs in terms of balanced hypergraph bicolorings, separators, and splitting sets. In turn, this provides complexity bounds for algebraic statistical models associated to hypergraphs. As two main applications, we recover a well-known complexity result for Markov bases of arbitrary 3-way tables, and we show that the defining ideal of the tangential variety i…