Block Stanley Decompositions II. Greedy Algorithms, Applications, and Open Problems

  • James Murdock
  • Theodore Murdock
Original Paper


Stanley decompositions are used in applied mathematics (dynamical systems) and \(\mathfrak {sl}_2\) invariant theory as finite descriptions of the set of standard monomials of a monomial ideal. The block notation for Stanley decompositions has proved itself in this context as a shorter notation and one that is useful in formulating algorithms such as the “box product”. Since the box product appears only in dynamical systems literature, we sketch its purpose and the role of block notation in this application. Then we present a greedy algorithm that produces incompressible block decompositions (called “organized”) from the monomial ideal; these are desirable for their likely brevity. Several open problems are proposed. We also continue to simplify the statement of the Soleyman–Jahan condition for a Stanley decomposition to be prime (come from a prime filtration) and for a block decomposition to be subprime, and present a greedy algorithm to produce “stacked decompositions”, which are subprime.


Geometry of monomial ideals Simplest Stanley decompositions Incompressible block decompositions Algorithms Organized decompositions Stacked decompositions Subalgebras Hilbert bases Algebraic relations Classical invariant theory Equivariants Normal forms for dynamical systems Prime filtrations Soleyman–Jahan condition Janet decompositions 


  1. 1.
    Anwar, I.: Janet’s algorithm. Bull. Math. Soc. Sci. Math. Roum. 51, 11–19 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Billera, L.J., Cushman, R., Sanders, J.A.: The Stanley decomposition of the harmonic oscillator. Nederl. Akad. Wetensch. Indag. Math. 50, 375–393 (1988)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Briand, E., Luque, J.-G., Thibon, J.-Y.: A complete set of covariants of the four qubit system. J. Phys. A 36, 9915–9927 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (1997)zbMATHGoogle Scholar
  5. 5.
    Cushman, R., Sanders, J.A.: Nilpotent normal form in dimension 4. In: Chow, S.-N., Hale, J.K. (eds.) Dynamics of Infinite Dimensional Systems. NATO ASI Series, vol. F37, pp. 61–66. Springer, Berlin (1987)CrossRefGoogle Scholar
  6. 6.
    Cushman, R., Sanders, J.A.: A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part. In: Stanton, D. (ed.) Invariant Theory and Tableaux, pp. 82–106. Springer, New York (1990)Google Scholar
  7. 7.
    Cushman, R., Sanders, J.A., White, N.: Normal form for the \((2;n)\)-nilpotent vector field, using invariant theory. Physica D 30, 399–412 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duval, A.M., Goeckner, B., Klivans, C.J., Martin, J.L.: A non-partitionable Cohen–Macauley simplicial complex. Adv. Math. 299, 381–395 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gachigua, G., Malonza, D., Sigey, J.: Normal form for systems with linear part \({N}\_{3(n)}\). Appl. Math. 3, 1641–1647 (2012). (ISSN 2152-7385 print, ISSN 2152-7393 online)CrossRefGoogle Scholar
  10. 10.
    Gachigua, G., Malonza, D., Sigey, J.: Ring of invariants systems with linear part \({N}\_{3^{(n)}}\). Am. Int. J. Contemp. Res. 2(11), 86–99 (2012)Google Scholar
  11. 11.
    Gatermann, K.: Computer algebra methods for equivariant dynamical systems. In: Lecture Notes in Mathematics, vol. 1728. Springer, New York (2000)Google Scholar
  12. 12.
    Ichim, B., Katthan, K., Moyano-Fernandez, J.J.: Stanley depth and the lcm-lattice. J. Comb. Theory Ser. A 150, 295–322 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Malonza, D.: Normal forms for coupled Takens–Bogdanov systems. J. Nonlinear Math. Phys. 11, 376–398 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    McLagan, D., Smith, G.: Uniform bounds on multigraded regularity. J. Algebra Geom. 14, 137–164 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Murdock, J.: On the structure of nilpotent normal form modules. J. Diff. Equ. 180, 198–237 (2002). (errata in Lemma 4: \(s\) should be the minimum weight of the two chain tops, not the minimum length of the chains; the transvectant is undefined, not zero, when \(i>s\))MathSciNetCrossRefGoogle Scholar
  16. 16.
    Murdock, J.: Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York (2003). (Lemma 6.4.3 is occasionally incorrect, so the method of Sect. 6.4 should be replaced by that of [18])CrossRefGoogle Scholar
  17. 17.
    Murdock, J.: Box products in nilpotent normal form theory: the factoring method. J. Diff. Equ. 260, 1010–1077 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Murdock, J., Malonza, D.: An improved theory of asymptotic unfoldings. J. Diff. Equ. 247, 685–709 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Murdock, J., Murdock, T.: Block Stanley decompositions I: the elementary and gnomon decompositions. J. Pure Appl. Algebra 219, 2189–2205 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Murdock, J., Sanders, J.A.: A new transvectant algorithm for nilpotent normal forms. J. Diff. Equ. 238, 234–256 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Plesken, W., Robertz, D.: Janet’s approach to presentations and resolutions for polynomials and linear PDEs. Arch. Math. 84, 22–37 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sanders, J., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Springer, New York (2007)zbMATHGoogle Scholar
  23. 23.
    Sanders, J.A.: Stanley decomposition of the joint covariants of three quadratics. Regular Chaotic Dyn. 12, 732–735 (2007)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schwarz, F.: Loewy decomposition of linear differential equations. Bull. Math. Sci. 3, 19–71 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Soleyman-Jahan, A.: Prime filtratins of monomial ideals and polarizations. J. Algebra 312, 1011–1032 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sturmfels, B., White, N.: Computing combinatorial decompositions of rings. Combinatorica 11, 275–293 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.AmesUSA

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