Thomas Decomposition of Algebraic and Differential Systems

  • Thomas Bächler
  • Vladimir Gerdt
  • Markus Lange-Hegermann
  • Daniel Robertz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6244)


In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new algorithm. For algebraic systems simplicity means triangularity, squarefreeness and non-vanishing initials. For differential systems the algorithm provides not only algebraic simplicity but also involutivity. The algorithm has been implemented in Maple.


Simple System Polynomial Ring Decomposition Algorithm Polynomial System Stream Cipher 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AH05]
    Apel, J., Hemmecke, R.: Detecting unnecessary reductions in an involutive basis computation. J. Symbolic Comput. 40(4-5), 1131–1149 (2005), MR MR2169107 (2006j:13026)Google Scholar
  2. [BC99]
    Buium, A., Cassidy, P.J.: Differential algebraic geometry and differential algebraic groups: from algebraic differential equations to diophantine geometry. In: [Kol99], pp. 567–636 (1999)Google Scholar
  3. [BCG+03]
    Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., Robertz, D.: The MAPLE Package Janet: I. Polynomial Systems. II. Linear Partial Differential Equations. In: Proc. 6th Int. Workshop on Computer Algebra in Scientific Computing, Passau, Germany, pp. 31–54 (2003),
  4. [BH04]
    Boulier, F., Hubert, E.: DIFFALG: description, help pages and examples of use, Symbolic Computation Group, University of Waterloo, Ontario, Canada (1996-2004),
  5. [BKRM01]
    Bouziane, D., Rody, A.K., Maârouf, H.: Unmixed-dimensional decomposition of a finitely generated perfect differential ideal. J. Symbolic Comput. 31(6), 631–649 (2001), MR MR1834002 (2002c:12007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BLH10]
    Bächler, T., Lange-Hegermann, M.: AlgebraicThomas and DifferentialThomas: Thomas decomposition for algebraic and differential systems (2008-2010),
  7. [BLOP95]
    Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Representation for the radical of a finitely generated differential ideal. In: ISSAC, pp. 158–166 (1995)Google Scholar
  8. [BLOP09]
    Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Computing representations for radicals of finitely generated differential ideals. Appl. Algebra Engrg. Comm. Comput. 20(1), 73–121 (2009), MR MR2496662 (2010c:12005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Bou07]
    Boulier, F.: Differential elimination and biological modelling, Gröbner bases in symbolic analysis. Radon Ser. Comput. Appl. Math. 2, 109–137 (2007), MR MR2394771 (2009f:12005)MathSciNetzbMATHGoogle Scholar
  10. [Bou09]
    Boulier, F.: BLAD: Bibliothèques lilloises d’algèbre différentielle (2004-2009),
  11. [CGL+07]
    Chen, C., Golubitsky, O., Lemaire, F., Maza, M.M., Pan, W.: Comprehensive triangular decomposition. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 73–101. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. [Del00]
    Dellière, S.: D.m. wang simple systems and dynamic constructible closure, Rapport de Recherche No. 2000–16 de l’Université de Limoges (2000)Google Scholar
  13. [Dio92]
    Diop, S.: On universal observability. In: Proc. 31st Conference on Decision and Control, Tucaon, Arizona (1992)Google Scholar
  14. [Duc00]
    Ducos, L.: Optimizations of the subresultant algorithm. J. Pure Appl. Algebra 145(2), 149–163 (2000), MR MR1733249 (2000m:68187)Google Scholar
  15. [GB98a]
    Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Math. Comput. Simulation 45(5-6), 519–541 (1998), Simplification of systems of algebraic and differential equations with applications. MR MR1627129 (99e:13033) Google Scholar
  16. [GB98b]
    Gerdt, V.P., Blinkov, Y.A.: Minimal involutive bases. Math. Comput. Simulation 45(5-6), 543–560 (1998), Simplification of systems of algebraic and differential equations with applications. MR MR1627130 (99e:13034) Google Scholar
  17. [Ger99]
    Gerdt, V.P.: Completion of linear differential systems to involution. In: Computer Algebra in Scientific Computing—CASC 1999, Munich, pp. 115–137. Springer, Berlin (1999), MR MR1729618 (2001d:12010)Google Scholar
  18. [Ger05]
    Gerdt, V.P.: Involutive algorithms for computing Gröbner bases. In: Computational Commutative and Non-Commutative Algebraic Geometry, NATO Sci. Ser. III Comput. Syst. Sci., vol. 196, pp. 199–225. IOS, Amsterdam (2005), MR MR2179201 (2007c:13040) Google Scholar
  19. [Ger08]
    Gerdt, V.P.: On decomposition of algebraic PDE systems into simple subsystems. Acta Appl. Math. 101(1-3), 39–51 (2008), MR MR2383543 (2009c:35003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [GY06]
    Gerdt, V.P., Yanovich, D.A.: Investigation of the effectiveness of involutive criteria for computing polynomial Janet bases. Programming and Computer Software 32(3), 134–138 (2006), MR MR2267374 (2007e:13036)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [GYB01]
    Gerdt, V.P., Yanovich, D.A., Blinkov, Y.A.: Fast search for the Janet divisor. Programming and Computer Software 27(1), 22–24 (2001), MR MR1867717MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Hab48]
    Habicht, W.: Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens. Comment. Math. Helv. 21, 99–116 (1948), MR MR0023796 (9,405f)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Hub03a]
    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), MR MR2043699 (2005c:13034)CrossRefGoogle Scholar
  24. [Hub03b]
    Hubert, E.: Notes on triangular sets and triangulation-decomposition algorithms. II. Differential systems. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 40–87. Springer, Heidelberg (2003), MR MR2043700 (2005c:13035) CrossRefGoogle Scholar
  25. [Jan29]
    Janet, M.: Leçons sur les systèmes des équationes aux dérivées partielles. In: Cahiers Scientifiques IV, Gauthiers-Villars, Paris (1929)Google Scholar
  26. [Kol73]
    Kolchin, E.R.: Differential algebra and algebraic groups. Pure and Applied Mathematics, vol. 54. Academic Press, New York (1973), MR MR0568864 (58 #27929)zbMATHGoogle Scholar
  27. [Kol99]
    Kolchin, E.R.: Selected works of Ellis Kolchin with commentary. American Mathematical Society, Providence (1999); Commentaries by Borel, A., Singer, M.F., Poizat, B., Buium, A., Cassidy, P.J. (eds.) with a preface by Hyman Bass, Buium and Cassidy. MR MR1677530 (2000g:01042) Google Scholar
  28. [LMX05]
    Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library in Maple. SIGSAM Bull. 39(3), 96–97 (2005)CrossRefzbMATHGoogle Scholar
  29. [LW99]
    Li, Z., Wang, D.: Coherent, regular and simple systems in zero decompositions of partial differential systems. System Science and Mathematical Sciences 12, 43–60 (1999)zbMATHGoogle Scholar
  30. [Mis93]
    Mishra, B.: Algorithmic algebra. Texts and Monographs in Computer Science. Springer, New York (1993), MR MR1239443 (94j:68127)CrossRefzbMATHGoogle Scholar
  31. [Riq10]
    Riquier, F.: Les systèmes d’équations aux dérivées partielles (1910)Google Scholar
  32. [Rit50]
    Ritt, J.F.: Differential Algebra. In: American Mathematical Society Colloquium Publications, vol. XXXIII. American Mathematical Society, New York (1950), MR MR0035763 (12,7c)Google Scholar
  33. [Ros59]
    Rosenfeld, A.: Specializations in differential algebra. Trans. Amer. Math. Soc. 90, 394–407 (1959), MR MR0107642 (21 #6367)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Sei10]
    Seiler, W.M.: Involution. In: Algorithms and Computation in Mathematics, vol. 24. Springer, Berlin (2010), The formal theory of differential equations and its applications in computer algebra. MR MR2573958 Google Scholar
  35. [sGH09]
    shan Gao, X., Huang, Z.: Efficient characteristic set algorithms for equation solving in finite fields and application in analysis of stream ciphers, Cryptology ePrint Archive, Report 2009/637 (2009),
  36. [Tho37]
    Thomas, J.M.: Differential systems, vol. XXI. AMS Colloquium Publications (1937)Google Scholar
  37. [Tho62]
    Thomas, J.M.: Systems and roots. The William Byrd Press, Inc., Richmond Virginia (1962)zbMATHGoogle Scholar
  38. [Wan98]
    Wang, D.: Decomposing polynomial systems into simple systems. J. Symbolic Comput. 25(3), 295–314 (1998), MR MR1615318 (99d:68130)MathSciNetCrossRefzbMATHGoogle Scholar
  39. [Wan01]
    Wang, D.: Elimination methods. In: Texts and Monographs in Symbolic Computation. Springer, Vienna (2001), MR MR1826878 (2002i:13040)Google Scholar
  40. [Wan03]
    Wang, D.: ε psilon: description, help pages and examples of use (2003),
  41. [Wan04]
    Wang, D.: Elimination practice. Imperial College Press, London (2004), Software tools and applications, With 1 CD-ROM (UNIX/LINUX, Windows). MR MR2050441 (2005a:68001) Google Scholar
  42. [Wu00]
    Wu, W.-T.: Mathematics mechanization. In: Mathematics and its Applications, vol. 489, Kluwer Academic Publishers Group, Dordrecht (2000), Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving. MR MR1834540 (2003a:01005) Google Scholar
  43. [Yap00]
    Yap, C.K.: Fundamental problems of algorithmic algebra. Oxford University Press, New York (2000), MR MR1740761 (2000m:12014)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Thomas Bächler
    • 1
  • Vladimir Gerdt
    • 2
  • Markus Lange-Hegermann
    • 1
  • Daniel Robertz
    • 1
  1. 1.Lehrstuhl B für MathematikRWTH-Aachen UniversityGermany
  2. 2.Joint Institute for Nuclear ResearchDubnaRussia

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