Modular Techniques for Noncommutative Gröbner Bases

Abstract

We extend modular techniques for computing Gröbner bases from the commutative setting to the vast class of noncommutative G-algebras. As in the commutative case, an effective verification test is only known to us in the graded case. In the general case, our algorithm is probabilistic in the sense that the resulting Gröbner basis can only be expected to generate the given ideal, with high probability. We have implemented our algorithm in the computer algebra system Singular and give timings to compare its performance with that of other instances of Buchberger’s algorithm, testing examples from D-module theory as well as classical benchmark examples. A particular feature of the modular algorithm is that it allows parallel runs.

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Notes

  1. 1.

    “!!” means double factorial.

  2. 2.

    We have to use a weighted cardinality count: when enlarging \({\mathcal {P}}\), the total weight of the elements already present must be strictly smaller than the total weight of the new elements. Otherwise, though highly unlikely in practical terms, it may happen that only unlucky primes are accumulated.

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Correspondence to Viktor Levandovskyy.

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Viktor Levandovskyy: Supported by the SFB-TRR 195 “Symbolic Tools in Mathematics and their Applications” of the German Research Foundation (DFG). Sharwan K. Tiwari: Supported by the German Academic Exchange Service (DAAD).

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Decker, W., Eder, C., Levandovskyy, V. et al. Modular Techniques for Noncommutative Gröbner Bases. Math.Comput.Sci. 14, 19–33 (2020). https://doi.org/10.1007/s11786-019-00412-9

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Keywords

  • Noncommutative Gröbner bases
  • G-algebras
  • PBW-algebras
  • Modular techniques

Mathematics Subject Classification

  • 13P10