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A Pommaret bases approach to the degree of a polynomial ideal

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Abstract

In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly the influence of each generator. Then we establish a new dimension depending Bézout bound for the degree and use it to obtain a dimension depending bound for the ideal membership problem.

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Notes

  1. Note that the Krull dimension corresponds to the dimension as affine and not as projective variety, although we work exclusively with homogeneous ideals. We stick with the affine picture to facilitate comparison with other results which are also based on the dimension as affine variety.

  2. Please note that despite the similarity in notation \(\deg {(\mathcal {I})}\) and \(\deg {(\mathcal {I},\prec )}\) refer to very different objects!

  3. We follow here the conventions of [16]. In [38], a convention is used which corresponds to reverting the order of the variables \(x_{1},\ldots ,x_{n}\). This implies e.g. that the class is defined as the minimum and not the maximum. Thus care must be taken when transferring results of [38] to the conventions used in this article.

  4. In [38, 39] also complementary decompositions, i.e. direct sum decompositions of the complement of \(\mathrm{LT}(\mathcal {I})\) are discussed and it is shown that any Pommaret basis induces one. Then one can write down an explicit formula for \(\mathrm{HF}_{\mathcal {I}}\) with a similar structure as (2). However, this only transforms the problem into understanding the precise relationship between the complementary decomposition and the Pommaret basis. While this is relatively simple with regard to, say, \(\dim (\mathcal {I})\) and \({{\mathrm{depth}}}(\mathcal {I})\) (see the corresponding results in [38, 39]), the situation becomes non-trivial for \(\deg (\mathcal {I})\).

  5. Quasi stable ideals are also known by many other names like weakly stable ideals, ideals of nested type or ideals of Borel type.

  6. By generic position, we mean after a linear change of variables from a Zariski open set, see [1] for more details.

  7. Although we are dealing with a homogeneous ideal, we will always work with the dimension as affine variety.

  8. An ideal is called unmixed if all its associated prime ideals have the same dimension.

  9. This example has been provided by David Masser (private communication).

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Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful comments which helped us to improve the manuscript. The third author received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. H2020-FETOPEN-2015-CSA 712689.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Bentolhoda Binaei & Amir Hashemi

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Werner M. Seiler

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  1. Bentolhoda Binaei
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  2. Amir Hashemi
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Correspondence to Amir Hashemi.

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Binaei, B., Hashemi, A. & Seiler, W.M. A Pommaret bases approach to the degree of a polynomial ideal. AAECC 29, 283–301 (2018). https://doi.org/10.1007/s00200-017-0342-y

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