Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\mathrm{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ memory, where $|\cdot |$ denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr\"obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.
翻译：以应用地貌数据分析为动力, 我们给出了一个高效的算法, 用于计算( 最小) $K[ x, y] $M$( M$ 美元) 的大型 $K[ y] $M$, 美元是一个字段。 该算法将一个由免费模块组成的短链组合 $X\xrightrow{f} Y\xrightrow{g}Z$( xrightrow{g} Z$), 这样的算法可以让 $M\ cong\ {g} /\ mathrm}$。 它运行在时间上 $O( X) 3}3 ⁇ {Y} 3 ⁇ 3} 3} 美元, 并需要$O( X) 2 ⁇ Y ⁇ 2 ⁇ 2$( $2) $( $) maine memor $, 美元表示排名。 。 根据我们的算法计算, $Metti 数字的宽度数字是很容易计算。 我们的方法基于简单的矩阵减缩数算算法的软件工具, 。
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