论文标题
Bigraded Betti数字的组合公式
A Combinatorial Formula for the Bigraded Betti Numbers
论文作者
论文摘要
已经表明，$ 1 $ - 参数持久性模块具有非常简单的分类，即有一个称为条形码的离散不变式，该条形码完全表征了$ 1 $ - 参数持久性模块，直到同构为同构。相比之下，卡尔森（Carlsson）和Zomorodian表明，$ n $ - 参数持久模块在$ n> 1 $时没有这样的“不错”分类；每个离散不变的不完整。尽管它们不完整，但离散的不变性仍然可以洞悉多参数持续模块的属性。 $ 2 $ - 参数持久模块的一个经过良好研究的离散不变是Bigraded Betti数字。通过交换代数技术，众所周知，可以通过简单的组合公式从$ M $内的某些曲折模块的条形码中回收$ 2 $ - 参数持久模块$ m $的大型贝蒂数字。我们提供了仅依赖基本线性代数的该公式的替代证明。
It has been shown that $1$-parameter persistence modules have a very simple classification, namely there is a discrete invariant called a barcode that completely characterizes $1$-parameter persistence modules up to isomorphism. In contrast, Carlsson and Zomorodian showed that $n$-parameter persistence modules have no such "nice" classification when $n>1$; every discrete invariant is incomplete. Despite their incompleteness, discrete invariants can still provide insight into the properties of multiparameter persistence modules. A well-studied discrete invariant for $2$-parameter persistence modules is the bigraded Betti numbers. Through commutative algebra techniques, it is known that the bigraded Betti numbers of a $2$-parameter persistence module $M$ can be recovered from the barcodes of certain zigzag modules within $M$ via a simple combinatorial formula. We present an alternate proof of this formula that relies only on basic linear algebra.
