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A dynamic F4 algorithm to compute Gröbner bases

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

The F4 algorithm re-imagines Buchberger’s algorithm as the row reduction of a Macaulay matrix: each row corresponds to a polynomial; each reduction of one row by another corresponds to one step of reducing an S-polynomial; and any row that completes reduction with a new pivot position corresponds to a new element of the basis. On the other hand, each column corresponds to a term, so that while it is common in linear algebra to exchange a matrix’s columns during row reduction, this has not been done in F4-style algorithms, as it runs the risk of producing an incorrect result. We show that it is possible to adapt an analogous, “dynamic” technique for Buchberger-style algorithms to F4-style algorithms, and we examine its behavior on some commonly referenced benchmark ideals.

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Notes

  1. In 2014, a colleague of the author tried to use the well-known program gfan to list all the Gröbner bases of the homogeneous Cyclic-5 ideal. The computer ran out of memory.

  2. In other words, \(Select\_row\) rejects orderings that do not preserve \(\mathrm {lt}\left( g\right)\) for all \(g\in G\). This “restricted” approach keeps with most past work, but [19] has recently shown that “unrestricted” algorithms have advantages.

  3. Originally, [4, 16] both proposed a “Hilbert heuristic” based on an ideal’s Hilbert data. Other heuristics have been proposed, usually based on invariants of an ideal: the present author described a “Betti heuristic” based on an ideal’s Betti numbers [22], while Langeloh has proposed variants on these heuristics [19]. These are of interest, but to keep the present investigation focused, we direct the reader to the references for further information.

  4. We generally accessed Singular via Sage [24].

  5. Eder suggested this idea to the author in 2017.

  6. A “bonus” to indeterminacy can occur when it leads to a surprisingly small basis, surprisingly quickly. Figures 1 and 2 above have gone through several iterations as we modified the program,. One of our biggest disappointments occurred when one revision of the code doubled the time to compute a Gröbner basis for Cyclic-8h, while the final basis remained roughly the same size.

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Acknowledgements

The author would like to thank Christian Eder, Gabriel Langeloh, and Teo Mora for thoughtful discussion on this topic. He would also like to acknowledge HPC at The University of Southern Mississippi, supported by the National Science Foundation under the Major Research Instrumentation (MRI) program via Grant # ACI 1626217, without which he would not have been able to complete Example 5.

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    John Perry

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Perry, J. A dynamic F4 algorithm to compute Gröbner bases. AAECC 31, 411–434 (2020). https://doi.org/10.1007/s00200-020-00450-y

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