AI 中文总结
本文提出计算坐标饱和理想局部v-数的新框架，推导森林图LSS理想局部v-数公式，证明多类图的二项边理想等满足v-数不超过其正则度的结论。
AI 中文摘要
本文引入了一种新框架，用于计算一类称为坐标饱和理想的特定局部v-数，这类理想包含与图相关的若干类Lovász-Saks-Schrijver（LSS）理想和（广义）二项边理想。对于森林图G，我们推导出所有d≥2时LSS理想$L_G^{\boldsymbol{K}}(d)$的局部v-数的显式公式，记为$\boldsymbol{v}_{\boldsymbol{p}_{\boldsymbol{\theta}}(G)}(L_G^{\boldsymbol{K}}(d))$，其中$\boldsymbol{K}$是代数闭域。由此我们证明$\boldsymbol{v}(L_G^{\boldsymbol{K}}(d)) \boldsymbol{\boldsymbol{reg}(R/L_G^{\boldsymbol{K}}(d))}$，其中$\boldsymbol{v}(L_G^{\boldsymbol{K}}(d))$和$\boldsymbol{reg}(R/L_G^{\boldsymbol{K}}(d))$分别表示$L_G^{\boldsymbol{K}}(d)$的v-数和$R/L_G^{\boldsymbol{K}}(d)$的Castelnuovo-Mumford正则度。此外，我们给出了$\boldsymbol{v}_{I_{K_{n}}}(L_{G}^{\boldsymbol{R}}(2))$的上界，其中$\boldsymbol{R}$是实数域。我们还提供了奇偶二项边理想的局部v-数的组合描述，记为$\boldsymbol{v}_{\boldsymbol{p}^{+}(G)}(\boldsymbol{\boldsymbol{I}}_{G})$，并作为应用证明了若干类非二部图满足$\boldsymbol{v}(\boldsymbol{\boldsymbol{I}}_G) \boldsymbol{\boldsymbol{reg}(R/\boldsymbol{\boldsymbol{I}}_G)}$。最后，我们证明了闭图G的所有二项边理想幂次$J_G^k$都满足$\boldsymbol{v}(J_G^k) \boldsymbol{\boldsymbol{reg}(R/J_G^k)}$。
英文摘要
In this paper, we introduce a new framework for computing certain localized $\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lovász-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\mathrm{v}$-number of the LSS ideal $L_G^{\mathbb{K}}(d)$, denoted by $\mathrm{v}_{\mathfrak{p}_{\emptyset}(G)}(L_G^{\mathbb{K}}(d))$, for all $d \geq 2$, where $\mathbb{K}$ is an algebraically closed field. As a consequence, we prove that $\mathrm{v}(L_G^{\mathbb{K}}(d)) \leq \mathrm{reg}(R/L_G^{\mathbb{K}}(d)),$ where $\mathrm{v}(L_G^{\mathbb{K}}(d))$ and $\mathrm{reg}(R/L_G^{\mathbb{K}}(d))$ denote the $\mathrm{v}$-number of $L_G^{\mathbb{K}}(d)$ and the Castelnuovo-Mumford regularity of $R/L_G^{\mathbb{K}}(d)$, respectively. Also, we give an upper bound for $\mathrm{v}_{I_{K_{n}}}(L_{G}^{\mathbb{R}}(2))$, where $\mathbb{R}$ is field of real numbers. Furthermore, we provide combinatorial descriptions of the localized $\mathrm{v}$-number of parity binomial edge ideals, denoted by $\mathrm{v}_{\mathfrak{p}^{+}(G)}(\mathcal{I}_{G})$, and, as an application, show that $\mathrm{v}(\mathcal{I}_G) \leq \mathrm{reg}(R/\mathcal{I}_G)$ for several classes of non-bipartite graphs. Finally, we prove that $\mathrm{v}(J_G^k)\leq \mathrm{reg}(R/J_G^k)$ for all powers of binomial edge ideals of closed graphs $G$.
Comments40 Pages. Comments are welcome!
