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- W2912734548 abstract "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 $Xxrightarrow{f} Y xrightarrow{g} Z$ such that $Mcong ker{g}/operatorname{im}{f}$. 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 Gröbner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of 2-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." @default.
- W2912734548 created "2019-02-21" @default.
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- W2912734548 date "2022-05-26" @default.
- W2912734548 modified "2023-10-18" @default.
- W2912734548 title "Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology" @default.
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- W2912734548 doi "https://doi.org/10.1137/20m1388425" @default.
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