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Comprehensive Gröbner basis theory for a parametric polynomial ideal and the associated completion algorithm

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Abstract

Gröbner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Gröbner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Gröbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a Gröbner basis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to Gröbner basis theory are reexamined and/or further developed for the parametric case: (i) Definition of a comprehensive Gröbner basis, (ii) test for a comprehensive Gröbner basis, (iii) parameterized rewriting, (iv) S-polynomials among parametric polynomials, (v) completion algorithm for directly computing a comprehensive Gröbner basis from a given basis of a parametric ideal. Elegant properties of Gröbner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique Gröbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Gröbner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.

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Acknowledgments

This paper is an expanded version of an invited talk at EACA, 2014 in Barcelona, in June 2014[19] and a paper in ISSAC 2015[20]. These papers result from a collaboration with Yiming Yang. He is not a coauthor of this paper as he chose to drop out of the project. The author would like to thank Prof. Yao Sun for many helpful discussions on this topic. The help provided by Prof. Montes through emails is appreciated.

Many of the concepts in this paper are closely related to similar concepts proposed in [1, 2].

Some of this work was done during the first author’s sabbatical at the Institute of Software, the Chinese Academy of Sciences in Beijing, China.

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Authors and Affiliations

  1. Department of Computer Science, University of New Mexico, Albuquerque, NM, USA

    Deepak Kapur

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  1. Deepak Kapur
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Correspondence to Deepak Kapur.

Additional information

This research was supported by the National Science Foundation under Grant No. DMS-1217054.

This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.

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Kapur, D. Comprehensive Gröbner basis theory for a parametric polynomial ideal and the associated completion algorithm. J Syst Sci Complex 30, 196–233 (2017). https://doi.org/10.1007/s11424-017-6337-8

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  • DOI: https://doi.org/10.1007/s11424-017-6337-8

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