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Computing Coupled Border Bases

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Abstract

Kehrein and Kreuzer (J Pure Appl Algebra 205(2):279–295, 2006) presented an algorithm, referred to as the BorderBasis algorithm, for computing border bases. The main objective of this paper is to improve this computation by applying new structures from the theory of Gröbner bases. In doing so, based on the method developed by Gao et al. (Math Comput 85(297):449–465, 2016), a new variant of the BorderBasis algorithm is proposed to compute simultaneously a border basis and also a Gröbner basis for the syzygy module of the input polynomials (this pair of bases will be called a coupled border basis). The new algorithm, and also the original form of the BorderBasis algorithm, have been implemented in the computer algebra systems Maple and ApCoCoA. We compare the performance of our implementation of these algorithms via a set of benchmark polynomials. Our experiments show that our algorithm performs more efficiently than the original BorderBasis algorithm.

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Notes

  1. By this, we mean that in the division process the leading terms are considered with respect to the coupled term ordering.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful and constructive comments.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Amir Hashemi & Samira Pourkhajouei

  2. Fakultät für Informatik und Mathematik, Universität Passau, Innst-rasse 33, 94032, Passau, Germany

    Martin Kreuzer

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  1. Amir Hashemi
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  2. Martin Kreuzer
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  3. Samira Pourkhajouei
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Correspondence to Amir Hashemi.

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Hashemi, A., Kreuzer, M. & Pourkhajouei, S. Computing Coupled Border Bases. Math.Comput.Sci. 14, 123–140 (2020). https://doi.org/10.1007/s11786-020-00452-6

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