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Gröbner Bases of Modules and Faugère’s \(F_4\) Algorithm in Isabelle/HOL

  • Alexander Maletzky
  • Fabian Immler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11006)

Abstract

We present an elegant, generic and extensive formalization of Gröbner bases, an important mathematical theory in the field of computer algebra, in Isabelle/HOL. The formalization covers all of the essentials of the theory (polynomial reduction, S-polynomials, Buchberger’s algorithm, Buchberger’s criteria for avoiding useless pairs), but also includes more advanced features like reduced Gröbner bases. Particular highlights are the first-time formalization of Faugère’s matrix-based \(F_4\) algorithm and the fact that the entire theory is formulated for modules and submodules rather than rings and ideals. All formalized algorithms can be translated into executable code operating on concrete data structures, enabling the certified computation of (reduced) Gröbner bases and syzygy modules.

Notes

Acknowledgments

We thank the anonymous referees for their valuable comments.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.RISC, Johannes Kepler Universität LinzLinzAustria
  2. 2.Institut für Informatik, Technische Universität MünchenMunichGermany

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