论文标题
降低边缘理想深度函数的行为
Decreasing behavior of the depth functions of edge ideals
论文作者
论文摘要
令$ i $为连接的非双方图和基本多项式环的$ r $的边缘理想。然后$ \ operatotorName {depth} r/i \ ge 1 $和$ \ operatatorName {depth} r/i^t = 0 $ for $ t \ gg 1 $。我们为$ \ operatatorName {depth} r/i^t = 1 $提供组合条件。尤其是，深度功能在达到1后迅速降至0。我们表明，如果$ \ operatatorName {depth} r/i = 1 $ = 1 $，则$ \ operatatOrname {depth} r/i^2 = 0 $，并且如果$ \ operatatorName {depth} r/i^2 = 1 $ then $ then $ then $ then $ \ then $ \ operatorAnnAmeameAmeAmeAmeAmeAmeAmeAmeAmeAmeAmeAmeAme {pepth} r/其他类似的结果表明，如果$ \ operatatorName {depth} r/i^t = 1 $，则$ \ operatatorName {depth} r/i^{t+3} = 0 $。这是一个令人惊讶的现象，因为功率的深度可以确定另一个力量的较小深度。此外，我们能够提供一个简单的组合标准，适用于$ \ operatatOrname {depth} r/i^{（t）} = 1 $ for $ t \ gg 1 $，并证明条件$ \ operatotorname {depth} r/i^{（t）} = 1 $ compisent， $ i $的权力。
Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for $\operatorname{depth} R/I^t = 1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\operatorname{depth} R/I = 1$ then $\operatorname{depth} R/I^2 = 0$ and if $\operatorname{depth} R/I^2 = 1$ then $\operatorname{depth} R/I^5 = 0$. Other similar results suggest that if $\operatorname{depth} R/I^t = 1$ then $\operatorname{depth} R/I^{t+3} = 0$. This a surprising phenomenon because the depth of a power can determine a smaller depth of another power. Furthermore, we are able to give a simple combinatorial criterion for $\operatorname{depth} R/I^{(t)} = 1$ for $t \gg 1$ and show that the condition $\operatorname{depth} R/I^{(t)} = 1$ is persistent, where $I^{(t)}$ denotes the $t$-th symbolic powers of $I$.
