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Motivated by applications to topological data analysis, we give an efficientalgorithm 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 complexof 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 thepresentation computed by our algorithm, the bigraded Betti numbers of $M$ arereadily 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 computingminimal presentations has been implemented in RIVET, a software tool for thevisualization and analysis of two-parameter persistent homology. In experimentson topological data analysis problems, our implementation outperforms thestandard computational commutative algebra packages Singular and Macaulay2 by awide margin.

Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology