Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ritt J, Differential Algebra, The American Mathematical Society, 1950.

    Book  MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  3. Wu W T, Basic principles of mechanical theorem proving in elementary geometries, Journal of Automated Reasoning, 1986, 2(3): 221–252.

    Article  MATH  Google Scholar 

  4. Wu W T, Mathematics Mechanization, Science Press/Kluwer, Beijing, 2001.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. Li X and Maza M M, Fast arithmetic for triangular sets: From theory to practice, International Symposium, 2007, 269–276.

    Google Scholar 

  11. Wang D, Elimination Methods, Springer, Vienna, 2001.

    Book  MATH  Google Scholar 

  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. Chen C, Davenport J H, May J P, et al., Triangular decomposition of semi-algebraic systems, Journal of Symbolic Computation, 2013, 49: 3–26.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  20. Hubert E, Factorization-free decomposition algorithms in differential algebra, Journal of Symbolic Computation, 2000, 29(4): 641–662.

    Article  MathSciNet  MATH  Google Scholar 

  21. Sit W Y, The Ritt-Kolchin theory for differential polynomials, Differential Algebra and Related Topics, 2002, 1–70.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  25. Gao X S, Huang Z, and Yuan C M, Binomial difference ideals, Journal of Symbolic Computation, 2017, 80: 665–706.

    Article  MathSciNet  Google Scholar 

  26. Gao X S, Huang Z, Wang J, et al., Toric difference variety, Journal of Systems Science and Complexity, 2017, 30(1): 173–195.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  31. Kolchin E R, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.

    MATH  Google Scholar 

  32. Rosenfeld A, Specializations in differential algebra, Transactions of the American Mathematical Society, 1959, 90(3): 394–407.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Youren Hu & Xiao-Shan Gao

  2. School of Mathematics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Youren Hu & Xiao-Shan Gao

Authors
  1. Youren Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao-Shan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Youren Hu.

Additional information

This research was supported by NKRDPC under Grant No. 2018YFA0306702 and the National Natural Science Foundation of China under Grant No. 11688101.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, Y., Gao, XS. Ritt-Wu Characteristic Set Method for Laurent Partial Differential Polynomial Systems. J Syst Sci Complex 32, 62–77 (2019). https://doi.org/10.1007/s11424-019-8343-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-019-8343-5

Keywords

Search

Navigation