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Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 62–77 | Cite as

Ritt-Wu Characteristic Set Method for Laurent Partial Differential Polynomial Systems

  • Youren HuEmail author
  • Xiao-Shan Gao
Article

Abstract

In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is defined and its basic properties are proved. The authors give a partial method to decide whether a Laurent differential chain A is Laurent regular. The decision for whether A is Laurent regular is reduced to the decision of whether a univariate differential chain A1 is Laurent regular. For a univariate differential chain A1, the authors first give a criterion for whether A1 is Laurent regular in terms of its generic zeros and then give partial results on deciding whether A1 is Laurent regular.

Keywords

Newton polygon Laurent partial differential polynomial system Laurent regular triangular set Ritt-Wu characteristic set 

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References

  1. [1]
    Ritt J, Differential Algebra, The American Mathematical Society, 1950.CrossRefzbMATHGoogle Scholar
  2. [2]
    Wu W T, On the decision problem and the mechanization of theorem-proving in elementary geometry, Science in China Ser. A, 1978, 29(2): 117–138.MathSciNetGoogle Scholar
  3. [3]
    Wu W T, Basic principles of mechanical theorem proving in elementary geometries, Journal of Automated Reasoning, 1986, 2(3): 221–252.CrossRefzbMATHGoogle Scholar
  4. [4]
    Wu W T, Mathematics Mechanization, Science Press/Kluwer, Beijing, 2001.Google Scholar
  5. [5]
    Aubry P, Lazard D, and Maza M M, On the theories of triangular sets, Journal of Symbolic Computation, 1999, 28(1–2): 105–124.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Chou S C and Gao X S, Ritt-Wu’s decomposition algorithm and geometry theorem proving, Tenth International Conference on Automated Deduction, Springer-Verlag, New York, 1990, 207–220.CrossRefGoogle Scholar
  7. [7]
    Cheng J S and Gao X S, Multiplicity-preserving triangular set decomposition of two polynomials, Journal of Systems Science and Complexity, 2014, 27(6): 1320–1344.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Li B H, A method to solve algebraic equations up to multiplicities via Ritt-Wu’s characteristic sets, Acta Analysis Functionalis Applicata, 2003, 5(2): 97–109.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Li B H, Hilbert problem 15 and Ritt-Wu method (I), Journal of Systems Science and Complexity, 2019, 32(1): 47–61.Google Scholar
  10. [10]
    Li X and Maza M M, Fast arithmetic for triangular sets: From theory to practice, International Symposium, 2007, 269–276.Google Scholar
  11. [11]
    Wang D, Elimination Methods, Springer, Vienna, 2001.CrossRefzbMATHGoogle Scholar
  12. [12]
    Yang L, Zhang J Z, and Hou X R, Non-Linear Algebraic Equations and Automated Theorem Proving, Shanghai Science and Education Pub., Shanghai, 1996.Google Scholar
  13. [13]
    Chen C, Davenport J H, May J P, et al., Triangular decomposition of semi-algebraic systems, Journal of Symbolic Computation, 2013, 49: 3–26.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Chai F, Gao X S, and Yuan C, A characteristic set method for solving boolean equations and applications in cryptanalysis of stream ciphers, Journal of Systems Science and Complexity, 2008, 21(2): 191–208.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Gao X S and Huang Z, Characteristic set algorithms for equation solving in finite fields, Journal of Symbolic Computation, 2011, 46(6): 655–679.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Li X, Mou C, and Wang D, Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case, Theoretical Computer Science, 2010, 60(11): 2983–2997.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Boulier F, Lemaire F, and Maza M M, Computing differential characteristic sets by change of ordering, Journal of Symbolic Computation, 2010, 45(1): 124–149.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Bouziane D, Rody A K, and Maârouf H, Unmixed-dimensional decomposition of a finitely generated perfect differential ideal, Journal of Symbolic Computation, 2001, 31(6): 631–649.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Chou S C and Gao X S, Automated reasonning in differential geometry and mechanics using the characteristic set method. Part I. An improved version of Ritt-Wu’s decomposition algorithm, Journal of Automated Reasonning, 1993, 10: 161–172.CrossRefzbMATHGoogle Scholar
  20. [20]
    Hubert E, Factorization-free decomposition algorithms in differential algebra, Journal of Symbolic Computation, 2000, 29(4): 641–662.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Sit W Y, The Ritt-Kolchin theory for differential polynomials, Differential Algebra and Related Topics, 2002, 1–70.Google Scholar
  22. [22]
    Zhu W and Gao X S, A triangular decomposition algorithm for differential polynomial systems with elementary computation complexity, Journal of Systems Science and Complexity, 2017, 30(2): 464–483.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Gao X S, Luo Y, and Yuan C M, A characteristic set method for ordinary difference polynomial systems, Journal of Symbolic Computation, 2009, 44(3): 242–260.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Wu W T and Gao X S, Mathematics mechanization and applications after thirty years, Frontiers of Computer Science in China, 2007, 1(1): 1–8.CrossRefGoogle Scholar
  25. [25]
    Gao X S, Huang Z, and Yuan C M, Binomial difference ideals, Journal of Symbolic Computation, 2017, 80: 665–706.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Gao X S, Huang Z, Wang J, et al., Toric difference variety, Journal of Systems Science and Complexity, 2017, 30(1): 173–195.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Hu Y and Gao X S, Characteristic set method for Laurent differential polynomial systems, International Workshop on Computer Algebra in Scientific Computing, Springer, 2017, 183–195.CrossRefGoogle Scholar
  28. [28]
    Cano J, An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms, Annales-Institut Fourier, 1993, 43(1): 125–142.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Fine H B, On the functions defined by differential equations, with an extension of the Puiseux polygon construction to these equations, American Journal of Mathematics, 1889, 11(4): 317–328.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Grigor’ev D Y and Singer M F, Solving ordinary differential equations in terms of series with real exponents, Transactions of the American Mathematical Society, 1991, 327(1): 329–351.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Kolchin E R, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.zbMATHGoogle Scholar
  32. [32]
    Rosenfeld A, Specializations in differential algebra, Transactions of the American Mathematical Society, 1959, 90(3): 394–407.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Li W and Li Y H, Computation of differential chow forms for ordinary prime differential ideals, Advances in Applied Mathematics, 2016, 72: 77–112.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.KLMM, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematics, University of Chinese Academy of SciencesChinese Academy of SciencesBeijingChina

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