On the Chordality of Simple Decomposition in Top-Down Style

  • Chenqi MouEmail author
  • Jiahua Lai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11989)


Simple decomposition of polynomial sets computes conditionally squarefree triangular sets or systems with certain zero or ideal relationships with the polynomial sets. In this paper we study the chordality of polynomial sets occurring in the process of simple decomposition in top-down style. We first reformulate Wang’s algorithm for simple decomposition in top-down style so that the decomposition process can be described in an inductive way. Then we prove that for a polynomial set whose associated graph is chordal, all the polynomial sets in the process of Wang’s algorithm for computing simple decomposition of this polynomial set have associated graphs which are subgraphs of the input chordal graph.


Chordal graph Simple decomposition Top-down style Triangular system 


  1. 1.
    Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symbolic Comput. 28(1–2), 105–124 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bächler, T.: Counting solutions of algebraic systems via triangular decomposition. Ph.D. thesis, RWTH Aachen University (2014)Google Scholar
  3. 3.
    Bächler, T., Gerdt, V., Lange-Hegermann, M., Robertz, D.: Algorithmic Thomas decomposition of algebraic and differential systems. J. Symbolic Comput. 47(10), 1233–1266 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.D. thesis, Universität Innsbruck, Austria (1965)Google Scholar
  5. 5.
    Chai, F., Gao, X.S., Yuan, C.: A characteristic set method for solving Boolean equations and applications in cryptanalysis of stream ciphers. J. Syst. Sci. Complex. 21(2), 191–208 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, C., Moreno Maza, M.: Algorithms for computing triangular decompositions of polynomial systems. J. Symbolic Comput. 47(6), 610–642 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cifuentes, D., Parrilo, P.A.: Chordal networks of polynomial ideals. SIAM J. Appl. Algebra Geom. 1(1), 73–110 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cox, D.A., Little, J.B., O’Shea, D.: Using Algebraic Geometry. Springer, Heidelberg (1998).
  9. 9.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner bases (\({F_4}\)). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)Google Scholar
  10. 10.
    Gao, X.S., Jiang, K.: Order in solving polynomial equations. In: Gao, X.S., Wang, D. (eds.) Computer Mathematics, Proceedings of ASCM 2000, pp. 308–318. World Scientific (2000)Google Scholar
  11. 11.
    Gao, X.S., Chou, S.C.: Solving parametric algebraic systems. In: Wang, P. (ed.) Proceedings of ISSAC 1992, pp. 335–341. ACM (1992)Google Scholar
  12. 12.
    Gerdt, V., Robertz, D.: Lagrangian constraints and differential Thomas decomposition. Adv. Appl. Math. 72, 113–138 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gilbert, J.R.: Predicting structure in sparse matrix computations. SIAM J. Matrix Anal. Appl. 15(1), 62–79 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huang, Z., Lin, D.: Attacking Bivium and Trivium with the characteristic set method. In: Nitaj, A., Pointcheval, D. (eds.) AFRICACRYPT 2011. LNCS, vol. 6737, pp. 77–91. Springer, Heidelberg (2011). Scholar
  15. 15.
    Hubert, E.: Notes on triangular sets and triangulation-decomposition algorithms i: polynomial systems. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 1–39. Springer, Heidelberg (2003). Scholar
  16. 16.
    Kalkbrener, M.: A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symbolic Comput. 15(2), 143–167 (1993)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mou, C., Wang, D., Li, X.: Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case. Theoret. Comput. Sci. 468, 102–113 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mou, C.: Symbolic detection of steady states of autonomous differential biological systems by transformation into block triangular form. In: Jansson, J., Martín-Vide, C., Vega-Rodríguez, M.A. (eds.) AlCoB 2018. LNCS, vol. 10849, pp. 115–127. Springer, Cham (2018). Scholar
  19. 19.
    Mou, C., Bai, Y.: On the chordality of polynomial sets in triangular decomposition in top-down style. In: Arreche, C. (ed.) Proceedings of ISSAC 2018, pp. 287–294. ACM (2018)Google Scholar
  20. 20.
    Mou, C., Bai, Y., Lai, J.: Chordal graphs in triangular decomposition in top-down style. J. Symbolic Comput. (2019, to appear)Google Scholar
  21. 21.
    Parter, S.: The use of linear graphs in Gauss elimination. SIAM Rev. 3(2), 119–130 (1961)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rose, D.J.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32(3), 597–609 (1970)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, D.: An elimination method for polynomial systems. J. Symbolic Comput. 16(2), 83–114 (1993)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, D.: Decomposing polynomial systems into simple systems. J. Symbolic Comput. 25(3), 295–314 (1998)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang, D.: Computing triangular systems and regular systems. J. Symbolic Comput. 30(2), 221–236 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, D.: Elimination Methods. Springer, Wien (2001). Scholar
  28. 28.
    Wu, W.T.: On zeros of algebraic equations: An application of Ritt principle. Kexue Tongbao 31(1), 1–5 (1986)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wu, W.T.: Mechanical Theorem Proving in Geometries: Basic Principles. Springer, Heidelberg (1994).
  30. 30.
    Yang, L., Zhang, J.Z.: Searching dependency between algebraic equations: An algorithm applied to automated reasoning. In: Johnson, J., McKee, S., Vella, A. (eds.) Artificial Intelligence in Mathematics, pp. 147–156. Oxford University Press, Oxford (1994)Google Scholar

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Authors and Affiliations

  1. 1.LMIB–School of Mathematical SciencesBeihang UniversityBeijingChina
  2. 2.Beijing Advanced Innovation Center for Big Data and Brain ComputingBeihang UniversityBeijingChina

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