A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition.

翻译：系数在域 $\mathbb{F}$ 上的持久性模 $M$ 是 $A_n$ 型等方向箭图的有限维线性表示，等价地，也是多项式环 $\mathbb{F}[x]$ 上的分次模。众所周知，$M$ 可分解为不可分解表示的直和，或由齐次元素生成的循环子模的直和。$M$ 的一个区间基是指 $M$ 中一组齐次元素，使得由它们生成的 $M$ 的循环子模之和是直和且等于 $M$。本文提出一种计算 $M$ 的区间基的新算法。该算法基于结构映射核的一个旗分解，适用于并行或分布式计算，且不依赖于 $M$ 的表示。该算法在性能上优于通过表示矩阵和史密斯标准形的传统方法。我们将并行方法专门应用于持久同调模，并通过将所提算法应用于基于霍奇分解的谐波追踪来结束讨论。