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


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.


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

Mathematics Subject Classification

13P10 13F20 14Q20 68W30 



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.


  1. 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)CrossRefGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bermejo, I., Gimenez, P.: Saturation and Castelnuovo–Mumford regularity. J. Algebra 303(2), 592–617 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cox, D.A., Little, J., O’Shea, D.: Using Algebraic Geometry, vol. 185, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  5. 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)Google Scholar
  6. 6.
    Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, Berlin (1995)zbMATHGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)Google Scholar
  9. 9.
    Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Math. Comput. Simul. 45(5–6), 519–541 (1998)MathSciNetCrossRefGoogle Scholar
  10. 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)MathSciNetCrossRefGoogle Scholar
  11. 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)Google Scholar
  12. 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)Google Scholar
  13. 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)CrossRefGoogle Scholar
  14. 14.
    Hashemi, A., Schweinfurter, M., Seiler, W.M.: Deterministic genericity for polynomial ideals. J. Symb. Comput. 86, 20–50 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hironaka, H.: Idealistic exponents of singularity. In: Algebraic Geometry—The Johns Hopkins Centennial Lectures, pp. 52–125. Johns Hopkins University Press, Baltimore (1977)Google Scholar
  16. 16.
    Janet, M.: Leçons sur les systèmes d’équations aux dérivées partielles. Fascicule IV. Gauthier-Villars, Cahiers Scientifiques, Paris (1929)zbMATHGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 18.
    Lecerf, G.: Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complex. 19(4), 564–596 (2003)MathSciNetCrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 20.
    Rees, D.: A basis theorem for polynomial modules. Proc. Camb. Philos. Soc. 52, 12–16 (1956)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Riquier, C.: Les systèmes d’équations aux derivées partielles. Gauthier-Villars, Paris (1910)zbMATHGoogle Scholar
  22. 22.
    Robertz, D.: Noether normalization guided by monomial cone decompositions. J. Symb. Comput. 44(10), 1359–1373 (2009)MathSciNetCrossRefGoogle Scholar
  23. 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. 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. 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)MathSciNetCrossRefGoogle Scholar
  26. 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. 27.
    Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sturmfels, B., White, N.: Computing combinatorial decompositions of rings. Combinatorica 11, 275–293 (1991)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zharkov, A.Yu., Blinkov, Y.A.: Involution approach to investigating polynomial systems. Math. Comput. Simul. 42(4–6):323–332 (1996)MathSciNetCrossRefGoogle Scholar

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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