发表机构
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文比较单项式理想及其整闭包的v-数，证明特定类单项式理想满足v-数不增的不等式，给出反例说明一般不成立，还得到连通图边理想幂次的v-数等式及不连通图的相关不等式。
AI 中文摘要
设$I$为标准分次多项式环中的单项式理想，$\u0304{I}$表示其整闭包。我们研究$\u0305{v}(I)$与$\u0305{v}(\u0304{I})$的关系。证明了二元变量单项式理想、三元变量等生成单项式理想及若干特殊类单项式理想满足$\u0305{v}(\u0304{I}) \u2264 \u0305{v}(I)$，同时给出反例表明该不等式一般不成立。对连通图$G$的边理想$I(G)$，证明当$k \u2265 1+|E(G)|$时，对所有$k$有$\u0305{v}(I(G)^k)=\u0305{v}(\u0304{I(G)^k}) = 2k-1$；当$G$不连通时，证明对所有足够大的$k$有$\u0305{v}(\u0304{I(G)^k}) \u2264 \u0305{v}(I(G)^k)$。
英文摘要
Let $I$ be a monomial ideal in a standard graded polynomial ring and let $\overline{I}$ denote its integral closure. We study the relationship between $\mathrm{v}(I)$ and $\mathrm{v}(\overline{I})$. We prove that $\mathrm{v}(\overline{I}) \leq \mathrm{v}(I)$ for monomial ideals in two variables, for equigenerated monomial ideals in three variables and for several special classes of monomial ideals, while providing examples showing that this inequality does not hold in general. For the edge ideal $I(G)$ of a connected graph $G$, we show that $\mathrm{v}(I(G)^k)=\mathrm{v}(\overline{I(G)^k}) = 2k-1$ for all $k \geq 1+|E(G)|$. Moreover, when $G$ is disconnected, we prove that $\mathrm{v}(\overline{I(G)^k})\leq\mathrm{v}({I(G)^k})$ for all sufficiently large $k$.
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