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Complexity of Membership Problems of Different Types of Polynomial Ideals

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Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

Abstract

We survey degree bounds and complexity classes of the word problem for polynomial ideals and related problems. The word problem for general polynomial ideals is known to be exponential space-complete, but there are several interesting subclasses of polynomial ideals that allow for better bounds. We review complexity results for polynomial ideals with low degree, toric ideals, binomial ideals, and radical ideals. Previously known results as well as recent findings in our project “Degree Bounds for Gröbner Bases of Important Classes of Polynomial Ideals and Efficient Algorithms” are presented.

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Acknowledgements

This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) through priority program SPP 1489 “Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory” in the project “Degree Bounds for Gröbner Bases of Important Classes of Polynomial Ideals and Efficient Algorithms”, TUM Graduate School, and TopMath, a graduate program of the Elite Network of Bavaria. We are grateful for their support.

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Authors and Affiliations

  1. Institut für Informatik, TU München, München, Germany

    Ernst W. Mayr & Stefan Toman

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  1. Ernst W. Mayr
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  2. Stefan Toman
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Correspondence to Stefan Toman .

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Editors and Affiliations

  1. IWR, Heidelberg University, Heidelberg, Germany

    Gebhard Böckle

  2. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Wolfram Decker

  3. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Gunter Malle

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Mayr, E.W., Toman, S. (2017). Complexity of Membership Problems of Different Types of Polynomial Ideals. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_20

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