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Degree Upper Bounds for Involutive Bases

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Abstract

The aim of this paper is to investigate upper bounds for the maximum degree of the elements of any minimal Janet basis of an ideal generated by a set of homogeneous polynomials. The presented bounds depend on the number of variables and the maximum degree of the generating set of the ideal. For this purpose, by giving a deeper analysis of the method due to Dubé (SIAM J Comput 19:750–773, 1990), we improve (and correct) his bound on the degrees of the elements of a reduced Gröbner basis. By giving a simple proof, it is shown that this new bound is valid for Pommaret bases, as well. Furthermore, based on Dubé’s method, and by introducing two new notions of genericity, so-called J-stable position and prime position, we show that Dubé’s (new) bound holds also for the maximum degree of polynomials in any minimal Janet basis of a homogeneous ideal in any of these positions. Finally, we study the introduced generic positions by proposing deterministic algorithms to transform any given homogeneous ideal into these positions.

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Notes

  1. The Maple code to calculate \({\mathcal {F}}_{1}(d)\) for any n is available at http://amirhashemi.iut.ac.ir/softwares

  2. The Maple code of our implementations are available at http://amirhashemi.iut.ac.ir/softwares

  3. For further details see the SymbolicData Project (http://www.SymbolicData.org)

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Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful and constructive comments. The research of the first author was in part supported by a Grant from IPM (No. 98550413).

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Authors and Affiliations

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

    Amir Hashemi & Hossein Parnian

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, P.O. Box: 19395-5746, Iran

    Amir Hashemi

  3. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Werner M. Seiler

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  1. Amir Hashemi
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  2. Hossein Parnian
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  3. Werner M. Seiler
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Correspondence to Amir Hashemi.

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Hashemi, A., Parnian, H. & Seiler, W.M. Degree Upper Bounds for Involutive Bases. Math.Comput.Sci. 15, 233–254 (2021). https://doi.org/10.1007/s11786-020-00480-2

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  • DOI: https://doi.org/10.1007/s11786-020-00480-2

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