Skip to main content
SpringerLink
Log in

Nœther Bases and Their Applications

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We refer to http://math.rwth-aachen.de/Janet/involutive. for the Maple implementation of this algorithm.

  2. The Maple code of our implementations and examples are available at http://amirhashemi.iut.ac.ir/softwares.

  3. For further details see the SymbolicData Project (http://www.SymbolicData.org).

References

  1. Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of the \(F_5\) Gröbner basis algorithm. J. Symb. Comput. 70, 49–70 (2015)

    Article  Google Scholar 

  2. Bermejo, I., Gimenez, P.: Computing the Castelnuovo–Mumford regularity of some subschemes of \(\mathbb{P}_K^n\) using quotients of monomial ideals. J. Pure Appl. Algebra 164(1–2), 23–33 (2001)

    Article  MathSciNet  Google Scholar 

  3. Bermejo, I., Gimenez, P.: Saturation and Castelnuovo–Mumford regularity. J. Algebra 303(2), 592–617 (2006)

    Article  MathSciNet  Google Scholar 

  4. Cox, D.A., Little, J., O’Shea, D.: Using Algebraic Geometry, vol. 185, 2nd edn. Springer, New York (2005)

    MATH  Google Scholar 

  5. Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 4th revised edn. Springer, Cham (2015)

  6. Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, Berlin (1995)

    MATH  Google Scholar 

  7. Gerdt, V.P.: On the relation between Pommaret and Janet bases. In: Computer Algebra in Scientific Computing (Samarkand, 2000), pp. 167–181. Springer, Berlin (2000)

    Chapter  Google Scholar 

  8. Gerdt, V.P.: Involutive algorithms for computing Gröbner bases. In: Computational Commutative and Non-commutative Algebraic Geometry. Proceedings of the NATO Advanced Research Workshop, 2004, pp. 199–225. IOS Press, Amsterdam (2005)

  9. Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Math. Comput. Simul. 45(5–6), 519–541 (1998)

    Article  MathSciNet  Google Scholar 

  10. Giusti, M., Hägele, K., Lecerf, G., Marchand, J., Salvy, B.: The projective Noether Maple package: computing the dimension of a projective variety. J. Symb. Comput. 30(3), 291–307 (2000)

    Article  MathSciNet  Google Scholar 

  11. Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra. With Contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, 2nd extended edn. Springer, Berlin (2007)

  12. Hashemi, A.: Efficient algorithms for computing Noether normalization. In: Asian Symposium on Computer Mathematics, Lecture Notes in Computer Science, vol. 5081, pp. 97–107. Springer, Berlin (2008)

  13. Hashemi, A.: Effective computation of radical of ideals and its application to invariant theory. In: International Congress on Mathematical Software, Lecture Notes in Computer Science, vol. 8592, pp. 382–389. Springer, Berlin (2014)

    Chapter  Google Scholar 

  14. Hashemi, A., Schweinfurter, M., Seiler, W.M.: Deterministic genericity for polynomial ideals. J. Symb. Comput. 86, 20–50 (2018)

    Article  MathSciNet  Google Scholar 

  15. Hironaka, H.: Idealistic exponents of singularity. In: Algebraic Geometry—The Johns Hopkins Centennial Lectures, pp. 52–125. Johns Hopkins University Press, Baltimore (1977)

  16. Janet, M.: Leçons sur les systèmes d’équations aux dérivées partielles. Fascicule IV. Gauthier-Villars, Cahiers Scientifiques, Paris (1929)

    MATH  Google Scholar 

  17. Krick, T., Logar, A.: An algorithm for the computation of the radical of an ideal in the ring of polynomials. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, vol. 539, pp. 195–205. Springer, Berlin (1991)

    Chapter  Google Scholar 

  18. Lecerf, G.: Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complex. 19(4), 564–596 (2003)

    Article  MathSciNet  Google Scholar 

  19. Logar, A.: A Computational Proof of the Noether Normalization Lemma. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 357, pp. 259–273 (1989)

    Chapter  Google Scholar 

  20. Rees, D.: A basis theorem for polynomial modules. Proc. Camb. Philos. Soc. 52, 12–16 (1956)

    Article  MathSciNet  Google Scholar 

  21. Riquier, C.: Les systèmes d’équations aux derivées partielles. Gauthier-Villars, Paris (1910)

    MATH  Google Scholar 

  22. Robertz, D.: Noether normalization guided by monomial cone decompositions. J. Symb. Comput. 44(10), 1359–1373 (2009)

    Article  MathSciNet  Google Scholar 

  23. Schreyer, F.-O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrass’schen Divisionssatz, Master’s thesis. University of Hamburg, Germany (1980)

    Google Scholar 

  24. Schweinfurter, M.: Deterministic Genericity and the Computation of Homological Invariants, Ph.D. thesis. Fachbereich Mathematik und Naturwissenschaften, Universität Kassel, Kassel (2016)

    Google Scholar 

  25. Seiler, W.M.: A combinatorial approach to involution and \(\delta \)-regularity. II. Structure analysis of polynomial modules with Pommaret bases. Appl. Algebra Eng. Commun. Comput. 20(3–4), 261–338 (2009)

    Article  MathSciNet  Google Scholar 

  26. Seiler, W.M.: Involution: The Formal Theory of Differential Equations and Its Applications in Computer Algebra. Algorithms and Computation in Mathematics. Springer, Berlin (2009)

    Google Scholar 

  27. Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)

    Article  MathSciNet  Google Scholar 

  28. Sturmfels, B., White, N.: Computing combinatorial decompositions of rings. Combinatorica 11, 275–293 (1991)

    Article  MathSciNet  Google Scholar 

  29. Zharkov, A.Yu., Blinkov, Y.A.: Involution approach to investigating polynomial systems. Math. Comput. Simul. 42(4–6):323–332 (1996)

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

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

    Amir Hashemi & Hossein Parnian

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

    Werner M. Seiler

Authors
  1. Amir Hashemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hossein Parnian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Werner M. Seiler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amir Hashemi.

Additional information

Communicated by Mohammad Taghi Dibaei.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hashemi, A., Parnian, H. & Seiler, W.M. Nœther Bases and Their Applications. Bull. Iran. Math. Soc. 45, 1283–1301 (2019). https://doi.org/10.1007/s41980-018-00198-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-018-00198-9

Keywords

Mathematics Subject Classification

Search

Navigation