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Relative Gröbner and Involutive Bases for Ideals in Quotient Rings


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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    We omit the index \({\mathcal {I}}\), if it is clear from the context by which ideal we factor.

  2. 2.

    It should be noted that, by Lemma 7.3, it follows that this change may be sparser than the change that we need to transform \({{\,\mathrm{{\mathcal {J}}}\,}}\) into quasi-stable position.


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The research of the first author was in part supported by a grant from IPM (No. 99130215). The third author thanks Cristina Bertone and Francesca Cioffi for their hospitality in Torino and for bringing this problem to his attention. We furthermore thank an anonymous referee for pointing out to us the work by Ceria and Mora on effectively given rings.

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Correspondence to Amir Hashemi.

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Hashemi, A., Orth, M. & Seiler, W.M. Relative Gröbner and Involutive Bases for Ideals in Quotient Rings. Math.Comput.Sci. 15, 453–482 (2021).

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  • Polynomial rings
  • Quotient rings
  • Ideals
  • Gröbner bases
  • Syzygy module
  • Involutive bases
  • Quasi-stable ideals

Mathematics Subject Classification

  • 13P10
  • 13D02
  • 68W30