Normal Forms for Operators via Gröbner Bases in Tensor Algebras

  • Jamal Hossein Poor
  • Clemens G. Raab
  • Georg Regensburger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9725)


We propose a general algorithmic approach to noncommutative operator algebras generated by linear operators using quotients of tensor algebras. In order to work with reduction systems in tensor algebras, Bergman’s setting provides a tensor analog of Gröbner bases. We discuss a modification of Bergman’s setting that allows for smaller reduction systems and tends to make computations more efficient. Verification of the confluence criterion based on S-polynomials has been implemented as a Mathematica package. Our implementation can also be used for computer-assisted construction of Gröbner bases starting from basic identities of operators. We illustrate our approach and the software using differential and integro-differential operators as examples.


Operator algebra Tensor algebra Noncommutative Gröbner basis Reduction systems 


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jamal Hossein Poor
    • 1
  • Clemens G. Raab
    • 1
  • Georg Regensburger
    • 1
  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of SciencesLinzAustria

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