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Comprehensive Gröbner Bases in a Java Computer Algebra System

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Computer Mathematics

Abstract

We present an implementation of the algorithms for computing comprehensive Gröbner bases in a Java computer algebra system (JAS). Contrary to approaches to implement comprehensive Gröbner bases with minimal requirements to the computer algebra system, we aim to provide all necessary algebraic structures occurring in the algorithm. In the implementation of a condition we aim at the maximal semantic exploitation of the occurring algebraic structures: the set of equations that equal zero are implemented as an ideal (with Gröbner base computation) and the set of inequalities are implemented as a multiplicative set which is simplified to polynomials of minimal degrees using squarefree or irreducible decomposition. The performance of our implementation is compared on well-known examples. With our approach we can also make the transition of a comprehensive Gröbner system to a polynomial ring over a regular coefficient ring and test or compute Gröbner bases.

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Acknowledgments

I thank Thomas Becker for discussions on the implementation of a polynomial template library and Raphael Jolly for the discussions on the generic type system suitable for a computer algebra system. JAS itself was improved by requirements from various users, especially Axel Kramer and by valuable feedback from other colleagues, in particular by Dongming Wang, Thomas Sturm, and Wolfgang K. Seiler, to name a few. This paper profited moreover from comments and feedback we received at the conference. Thanks also to Markus Aleksy and Hans-Günther Kruse for encouraging and supporting this work.

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Authors and Affiliations

  1. IT-Center, University of Mannheim, 68131, Mannheim, Germany

    Heinz Kredel

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  1. Heinz Kredel
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Correspondence to Heinz Kredel .

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Editors and Affiliations

  1. Academy of Mathematics & System Science, Beijing, China

    Ruyong Feng

  2. Department of Mathematics and Computer S, University of Antwerp, Antwerp, Antwerpen, Belgium

    Wen-shin Lee

  3. Department of Mathematical Information S, Tokyo University of Science, Tokyo, Tokyo, Japan

    Yosuke Sato

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Kredel, H. (2014). Comprehensive Gröbner Bases in a Java Computer Algebra System. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_9

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