Vertex cover algebras of unimodular hypergraphs

Regularity of powers of cover ideals of unimodular hypergraphs

Let $\H$ be a unimodular hypergraph over the vertex set $[n]$ and let $J(\H)$ be the cover ideal of $\H$ in the polynomial ring $R=K[x_1,\ldots,x_n]$. We show that $\reg J(\H)^s$ is a linear function in $s$ for all $s\geqslant r\left\lceil \frac{n}{2}\right\rceil+1$ where $r$ is the rank of $\H$. Moreover for every $i$, $a_i(R/J(\H)^s)$ is also a linear function in $s$ for $s \geqslant n^2$.

单模超图覆盖理想幂的V数

V-numbers of powers of cover ideals of unimodular hypergraphs

该研究证明了单模超图覆盖理想幂的局部v数、特定范围的全局v数，以及任意树覆盖理想幂的全局v数均随幂次t呈线性变化。