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Formalization of Dubé’s Degree Bounds for Gröbner Bases in Isabelle/HOL

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Intelligent Computer Mathematics (CICM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11617))

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Abstract

We present an Isabelle/HOL formalization of certain upper bounds on the degrees of Gröbner bases in multivariate polynomial rings over fields, due to Dubé. These bounds are not only of theoretical interest, but can also be used for computing Gröbner bases by row-reducing Macaulay matrices.

The formalization covers the whole theory developed by Dubé for obtaining the bounds, building upon an extensive existing library of multivariate polynomials and Gröbner bases in Isabelle/HOL. To the best of our knowledge, this is the first thorough formalization of degree bounds for Gröbner bases in any proof assistant.

The research was funded by the Austrian Science Fund (FWF): P 29498-N31.

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Notes

  1. 1.

    http://isabelle.in.tum.de/repos/isabelle.

  2. 2.

    http://devel.isa-afp.org/.

  3. 3.

    Reduced Gröbner bases are unique for every admissible order relation, but different orders may yield different reduced Gröbner bases for the same ideal.

  4. 4.

    \(\texttt {poly\_deg}~0\) is defined to be 0.

  5. 5.

    Of course, splits_wrt is defined for lists ps, qs instead of sets P, Q, but informally it is easier to think of sets.

  6. 6.

    Although asymptotically \(\mathrm {Dube}_{n,d}\) also has to grow double exponentially in n, as shown in [10].

References

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  7. Maletzky, A.: Gröbner bases and Macaulay matrices in Isabelle/HOL. Technical report, RISC, Johannes Kepler University Linz, Austria (2018). http://www.risc.jku.at/publications/download/risc_5814/Paper.pdf

  8. Maletzky, A.: Isabelle/HOL formalization of advanced Gröbner bases material (2019). https://github.com/amaletzk/Isabelle-Groebner/

  9. Maletzky, A., Immler, F.: Gröbner bases of modules and Faugère’s \(F_4\) algorithm in Isabelle/HOL. In: Rabe, F., Farmer, W.M., Passmore, G.O., Youssef, A. (eds.) CICM 2018. LNCS (LNAI), vol. 11006, pp. 178–193. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96812-4_16

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Acknowledgments

I thank the anonymous referees for their valuable comments.

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Authors and Affiliations

  1. RISC, Johannes Kepler University Linz, Linz, Austria

    Alexander Maletzky

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  1. Alexander Maletzky
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Correspondence to Alexander Maletzky .

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Editors and Affiliations

  1. University of Innsbruck, Innsbruck, Austria

    Cezary Kaliszyk

  2. University of St. Andrews, St. Andrews, UK

    Edwin Brady

  3. University of Applied Sciences, Neu-Ulm, Germany

    Andrea Kohlhase

  4. University of Bologna, Bologna, Italy

    Claudio Sacerdoti Coen

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Maletzky, A. (2019). Formalization of Dubé’s Degree Bounds for Gröbner Bases in Isabelle/HOL. In: Kaliszyk, C., Brady, E., Kohlhase, A., Sacerdoti Coen, C. (eds) Intelligent Computer Mathematics. CICM 2019. Lecture Notes in Computer Science(), vol 11617. Springer, Cham. https://doi.org/10.1007/978-3-030-23250-4_11

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  • DOI: https://doi.org/10.1007/978-3-030-23250-4_11

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  • Online ISBN: 978-3-030-23250-4

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