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Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 5, pp 1283–1301 | Cite as

Nœther Bases and Their Applications

  • Amir HashemiEmail author
  • Hossein Parnian
  • Werner M. Seiler
Original Paper

Abstract

In this paper, we introduce a new involutive division, called D-Nœther division, and the corresponding notion of a Nœther basis. It is shown that an ideal is in Nœther position, if and only if it possesses a finite Nœther basis. We present a deterministic algorithm which, given a homogeneous ideal, finds a linear change of variables so that the ideal after performing this change possesses a finite Nœther basis (and equivalently is in Nœther position). Furthermore, we define the new concept of an ideal of Nœther type and study its connections with Rees decompositions. We have implemented all the algorithms described in this paper in Maple and assess their performance on a number of benchmark examples.

Keywords

Polynomial ring Gröbner bases Involutive bases Pommaret bases Quasi-stable ideals Nœther position 

Mathematics Subject Classification

13P10 13F20 14Q20 68W30 

Notes

Acknowledgements

The work of Hossein Parnian was partially performed as part of the H2020-FETOPEN-2016-2017-CSA project \(SC^{2}\) (712689). The authors would like to thank the anonymous reviewers for their helpful comments.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  2. 2.Institut für MathematikUniversität KasselKasselGermany

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