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Paper   IPM / M / 16687
School of Mathematics
  Title:   Relative Grobner and involutive bases for ideals in quotient rings
  Author(s):  Amir Hashemi (Joint with M. Orht and W. M. Seiler)
  Status:   Published
  Journal: Math. Comput. Sci.
  Vol.:  15
  Year:  2021
  Pages:   453-482
  Supported by:  IPM
We extend the concept of Gröbner bases to relative Gröbner bases for ideals in and modules over quotient rings of a polynomial ring over a field. We develop a “relative” variant of both Buchberger’s criteria for avoiding reductions to zero and Schreyer’s theorem for a Gröbner basis of the syzygy module. As main contribution, we then introduce the novel notion of relative involutive bases and present an algorithm for their explicit construction. Finally, we define the new notion of relatively quasi-stable ideals and exploit it for the algorithmic determination of coordinates in which finite relative Pommaret bases exist.

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