Advertisement

Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 287–316 | Cite as

Elimination Theory in Differential and Difference Algebra

  • Wei LiEmail author
  • Chun-Ming Yuan
Article

Abstract

Elimination theory is central in differential and difference algebra. The Wu-Ritt characteristic set method, the resultant and the Chow form are three fundamental tools in the elimination theory for algebraic differential or difference equations. In this paper, the authors mainly present a survey of the existing work on the theory of characteristic set methods for differential and difference systems, the theory of differential Chow forms, and the theory of sparse differential and difference resultants.

Keywords

Differential Chow forms differential resultants sparse differential resultants Wu-Ritt characteristic sets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ritt J F, Differential Algebra, American Mathematical Society Colloquium Publications, Vol. XXXIII, American Mathematical Society, New York, 1950.Google Scholar
  2. [2]
    Kolchin E R, Differential Algebra and Algebraic Groups, Academic Press, New York-London, 1973.zbMATHGoogle Scholar
  3. [3]
    Cohn R M, Difference Algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965.zbMATHGoogle Scholar
  4. [4]
    Wu W T, On the decision problem and the mechanization of theorem-proving in elementary geometry, Sci. Sinica, 1978, 21(2): 159–172.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Wu W T, A constructive theory of differential algebraic geometry based on works of Ritt J F with particular applications to mechanical theorem-proving of differential geometries, Differential Geometry and Differential Equations (Shanghai, 1985), volume 1255 of Lecture Notes in Math., Springer, Berlin, 1987, 173–189.Google Scholar
  6. [6]
    Wu W T, Basic Principles of Mechanical Theorem Proving in Elementary Geometries, Science Press, Beijing, 1984; English translation, Springer, Wien, 1994.Google Scholar
  7. [7]
    Aubry P, Lazard D, and Moreno Maza M, On the theories of triangular sets, J. Symbolic Comput., 1999, 28(1): 105–124.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Boulier F, Lazard D, Ollivier F, et al., Representation for the radical of a finitely generated differential ideal, Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, ISSAC’95, Montreal, Canada, July 10–12, 1995, 158–166. ACM Press, New York, NY, 1995.Google Scholar
  9. [9]
    Bouziane D, Kandri Rody A, and Maârouf H, Unmixed-dimensional decomposition of a finitely generated perfect differential ideal, J. Symbolic Comput., 2001, 31(6): 631–649.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Hubert E, Factorization-free decomposition algorithms in differential algebra, J. Symbolic Comput., 2000, 29(4–5): 641–662.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Wu W T, Mathematics Machenization, Science Press/Kluwer, Beijing, 2001.Google Scholar
  12. [12]
    Yang L, Zhang J Z, and Hou X R, Non-linear Algebraic Equations and Automated Theorem Proving, Shanghai Science and Technological Education Pub., 1996 (in Chinese).Google Scholar
  13. [13]
    Ritt J F and Doob J L, Systems of algebraic difference equations, Amer. J. Math., 1933, 55(1–4): 505–514.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Ritt J F and Raudenbush H W, Ideal theory and algebraic difference equations, Trans. Amer. Math. Soc., 1939, 46: 445–452.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Gao X S, Luo Y, and Yuan C M, A characteristic set method for ordinary difference polynomial systems, J. Symbolic Comput., 2009, 44(3): 242–260.MathSciNetzbMATHGoogle Scholar
  16. [16]
    Gao X S and Yuan C M, Resolvent systems of difference polynomial ideals, ISSAC 2006, ACM, New York, 2006, 100–108.Google Scholar
  17. [17]
    Gao X S, Yuan C M, and Zhang G L, Ritt-Wu’s characteristic set method for ordinary difference polynomial systems with arbitrary ordering, Acta Math. Sci. Ser. B (Engl. Ed.), 2009, 29(4): 1063–1080.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Zhang G L and Gao X S. Properties of ascending chains for partial difference polynomial systems, Computer Mathematics 8th Asian Symposium, ASCM 2007, Singapore, December 15–17, 2007. Revised and invited papers, 307–321. Springer, Berlin, 2008.Google Scholar
  19. [19]
    Kondratieva M V, Levin A B, Mikhalev A V, et al., Differential and Difference Dimension Polynomials, volume 461 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1999.zbMATHGoogle Scholar
  20. [20]
    Gao X S, Van der Hoeven J, Yuan C M, et al., Characteristic set method for differential-difference polynomial systems, J. Symbolic Comput., 2009, 44(9): 1137–1163.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Gel’fand I M, Kapranov M M, and Zelevinsky A V, Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994.Google Scholar
  22. [22]
    Hodge W V D and Pedoe D, Methods of Algebraic Geometry, Vol. II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994.Google Scholar
  23. [23]
    Brownawell W D, Bounds for the degrees in the nullstellensatz, Ann. Math., 1987, 126(3): 577–591.MathSciNetzbMATHGoogle Scholar
  24. [24]
    Sturmfels B, Sparse elimination theory, Computational Algebraic Geometry and Commutative Algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge University Press, Cambridge, 1993, 264–298.Google Scholar
  25. [25]
    Nesterenko Y V, Estimates for the orders of zeros of functions of a certain class and applications in the theory of transcendental numbers, Izv. Akad. Nauk SSSR Ser. Mat., 1977, 41: 253–284.MathSciNetzbMATHGoogle Scholar
  26. [26]
    Philippon P, Critères pour l’indpendance algbrique, Inst. Hautes Ètudes Sci. Publ. Math., 1986, 64: 5–52.Google Scholar
  27. [27]
    Jeronimo G, Krick T, Sabia J, et al., The computational complexity of the Chow form, Found. Comput. Math., 2004, 4(1): 41–117.MathSciNetzbMATHGoogle Scholar
  28. [28]
    Gao X S, Li W, and Yuan C M, Intersection theory in differential algebraic geometry: Generic intersections and the differential Chow form, Trans. Amer. Math. Soc., 2013, 365(9): 4575–4632.MathSciNetzbMATHGoogle Scholar
  29. [29]
    Li W and Gao X S, Chow form for projective differential variety, J. Algebra, 2012, 370: 344–360.MathSciNetzbMATHGoogle Scholar
  30. [30]
    Freitag J, Li W, and Scanlon T, Differential chow varieties exist, J. Lond. Math. Soc., 2017, 95(1): 128–156.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Li W and Li Y H, Difference Chow form, J. Algebra, 2015, 428: 67–90.MathSciNetzbMATHGoogle Scholar
  32. [32]
    Li W, Partial differential chow forms and a type of partial differential chow varieties, ArXiv: 1709.02358v1, 2017.Google Scholar
  33. [33]
    Macaulay F S, The Algebraic Theory of Modular Systems, Cambridge University Press, Cambridge, 1994.zbMATHGoogle Scholar
  34. [34]
    Eisenbud D, Schreyer F, and Weyman J, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc., 2003, 16(3): 537–579.MathSciNetzbMATHGoogle Scholar
  35. [35]
    Jouanolou J P, Le formalisme du résultant, Adv. Math., 1991, 90(2): 117–263.MathSciNetzbMATHGoogle Scholar
  36. [36]
    Canny J, Generalised characteristic polynomials, J. Symb. Comput., 1990, 9(3): 241–250.MathSciNetzbMATHGoogle Scholar
  37. [37]
    Emiris I Z and Canny J F, Efficient incremental algorithms for the sparse resultant and the mixed volume, J. Symb. Comput., 1995, 20(2): 117–149.MathSciNetzbMATHGoogle Scholar
  38. [38]
    Emiris I Z and Pan V Y, Improved algorithms for computing determinants and resultants, J. Complexity, 2005, 21(1): 43–71.MathSciNetzbMATHGoogle Scholar
  39. [39]
    Bernstein D N, The number of roots of a system of equations, Functional Anal. Appl., 1975, 9(3): 183–185.MathSciNetzbMATHGoogle Scholar
  40. [40]
    Sturmfels B, On the Newton polytope of the resultant, J. Algebraic Combin., 1994, 3(2): 207–236.MathSciNetzbMATHGoogle Scholar
  41. [41]
    Emiris I Z, On the complexity of sparse elimination, J. Complexity, 1996, 12(2): 134–166.MathSciNetzbMATHGoogle Scholar
  42. [42]
    D’Andrea C, Macaulay style formulas for sparse resultants, Trans. Amer. Math. Soc., 2002, 354(7): 2595–2629.MathSciNetzbMATHGoogle Scholar
  43. [43]
    Ore O, Formale theorie der linearen differentialgleichungen (Zweiter Teil), J. Reine Angew. Math., 1932, 168: 233–252.MathSciNetzbMATHGoogle Scholar
  44. [44]
    Berkovich L M and Tsirulik V G, Differential resultants and some of their applications, Differ. Equations, 1986, 22: 530–536.zbMATHGoogle Scholar
  45. [45]
    Zeilberger D, A holonomic systems approach to special functions identities, J. Comput. Appl. Math., 1990, 32(3): 321–368.MathSciNetzbMATHGoogle Scholar
  46. [46]
    Chyzak F and Salvy B, Non-commutative elimination in Ore algebras proves multivariate identities, J. Symbolic Comput., 1998, 26(2): 187–227.MathSciNetzbMATHGoogle Scholar
  47. [47]
    Carrà Ferro G, A resultant theory for systems of linear partial differential equations, Lie Groups Appl., 1994, 1(1): 47–55.MathSciNetzbMATHGoogle Scholar
  48. [48]
    Chardin M, Differential resultants and subresultants, Fundamentals of computation theory (Gosen, 1991), volume 529 of Lecture Notes in Comput. Sci., Springer, Berlin, 1991, 180–189.zbMATHGoogle Scholar
  49. [49]
    Li Z M, A subresultant theory for linear differential, linear difference and Ore polynomials, with applications, PhD thesis, Johannes Kepler University, 1996.Google Scholar
  50. [50]
    Hong H, Ore subresultant coefficients in solutions, Appl. Algebra Engrg. Comm. Comput., 2001, 12(5): 421–428.MathSciNetzbMATHGoogle Scholar
  51. [51]
    Ritt J F, Differential Equations from the Algebraic Standpoint, American Mathematical Society, New York, 1932.zbMATHGoogle Scholar
  52. [52]
    Zwillinger D, Handbook of Differential Equations, Academic Press, San Diego, CA, 3rd Ed., 1998.zbMATHGoogle Scholar
  53. [53]
    Rueda S L and Sendra J R, Linear complete differential resultants and the implicitization of linear DPPEs, J. Symbolic Comput., 2010, 45(3): 324–341.MathSciNetzbMATHGoogle Scholar
  54. [54]
    Carrà-Ferro G, A resultant theory for the systems of two ordinary algebraic differential equations, Appl. Algebra Engrg. Comm. Comput., 1997, 8(6): 539–560.MathSciNetzbMATHGoogle Scholar
  55. [55]
    Li W, Yuan C M, and Gao X S, Sparse differential resultant for Laurent differential polynomials, Found. Comput. Math., 2015, 15(2): 451–517.MathSciNetzbMATHGoogle Scholar
  56. [56]
    Zhang Z Y, Yuan C M, and Gao X S, Matrix formulae of differential resultant for first order generic ordinary differential polynomials, Computer Mathematics 9th Asian Symposium, ASCM 2009, Fukuoka, Japan, December 14–17, 2009, 10th Asian symposium, ASCM 2012, Beijing, China, October 26–28, 2012, Contributed papers and invited talks, Springer, Berlin, 2014, 479–503.Google Scholar
  57. [57]
    Rueda S L, Linear sparse differential resultant formulas, Linear Algebra Appl., 2013, 438(11): 4296–4321.MathSciNetzbMATHGoogle Scholar
  58. [58]
    Li W, Yuan C M, and Gao X S. Sparse difference resultant, ISSAC 2013 — Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2013, 275–282.Google Scholar
  59. [59]
    Li W, Yuan C M, and Gao X S, Sparse difference resultant, J. Symbolic Comput., 2015, 68(1): 169–203.MathSciNetzbMATHGoogle Scholar
  60. [60]
    Yuan C M and Zhang Z Y, New bounds and efficient algorithm for sparse difference resultant, arXiv: 1810.00057, 2018.Google Scholar
  61. [61]
    Golubitsky O, Kondratieva M, Ovchinnikov A, et al., A bound for orders in differential Nullstellensatz, J. Algebra, 2009, 322(11): 3852–3877.MathSciNetzbMATHGoogle Scholar
  62. [62]
    D’Alfonso L, Jeronimo G, and Solernó P, Effective differential Nullstellensatz for ordinary DAE systems with constant coefficients, J. Complexity, 2014, 30(5): 588–603.MathSciNetzbMATHGoogle Scholar
  63. [63]
    Gustavson R, Kondratieva M, and Ovchinnikov A, New effective differential Nullstellensatz, Adv. Math., 2016, 290: 1138–1158.MathSciNetzbMATHGoogle Scholar
  64. [64]
    Ovchinnikov A, Pogudin G, and Scanlon T, Effective difference elimination and Nullstellensatz, ArXiv: 1712.01412V2, 2017.Google Scholar
  65. [65]
    Ovchinnikov A, Pogudin G, and Vo T N, Bounds for elimination of unknowns in systems of differential-algebraic equations, ArXiv: 1610.0422v6, 2018.Google Scholar
  66. [66]
    Yang L, Zeng Z B, and Zhang W N, Search dependency between algebraic equations: An algorithm applied to automated reasoning, Technical Report ICTP/91/6, International Center For Theoretical Physics, International Atomic Energy Agency, Miramare, Trieste, 1991.Google Scholar
  67. [67]
    Kalkbrener M, A generalized Euclidean algorithm for computing triangular representations of algebraic varieties, J. Symbolic Comput., 1993, 15(2): 143–167.MathSciNetzbMATHGoogle Scholar
  68. [68]
    Gallo G and Mishra B, Efficient algorithms and bounds for Wu-Ritt characteristic sets, Effective Methods in Algebraic Geometry (Castiglioncello, 1990), volume 94 of Progr. Math., Birkhäuser Boston, Boston, MA, 1991, 119–142.Google Scholar
  69. [69]
    Wang D M, An elimination method for polynomial systems, J. Symbolic Comput., 1993, 16(2): 83–114.MathSciNetzbMATHGoogle Scholar
  70. [70]
    Wang D M, Decomposing polynomial systems into simple systems, J. Symbolic Comput., 1998, 25(3): 295–314.MathSciNetzbMATHGoogle Scholar
  71. [71]
    Wang D M, Computing triangular systems and regular systems, J. Symbolic Comput., 2000, 30(2): 221–236.MathSciNetzbMATHGoogle Scholar
  72. [72]
    Hubert E, Notes on triangular sets and triangulation-decomposition algorithms. I. Polynomial systems, Symbolic and Numerical Scientific Computation (Hagenberg, 2001), volume 2630 of Lecture Notes in Comput. Sci., Springer, Berlin, 2003, 1–39.Google Scholar
  73. [73]
    Li Z M and Wang D M, Some properties of triangular sets and improvement upon algorithm charser, Eds. by Calmet J, Ida T, Wang D, Artificial Intelligence and Symbolic Computation, Lecture Notes of Comput. Sci., 2006, 4120: 82–93.zbMATHGoogle Scholar
  74. [74]
    Chou S C, Mechanical Geometry Theorem Proving, volume 41 of Mathematics and Its Applications, D. Reidel Publishing Co., Dordrecht, 1988.zbMATHGoogle Scholar
  75. [75]
    Chou S C and Gao X S, Ritt-Wu’s decomposition algorithm and geometry theorem proving, 10th International Conference on Automated Deduction (Kaiserslautern, 1990), volume 449 of Lecture Notes in Comput. Sci., 207–220, Springer, Berlin, 1990.Google Scholar
  76. [76]
    Chen C B, Davenport J H, May J P, et al., Triangular decomposition of semi-algebraic systems, J. Symb. Comput., 2013, 49: 3–26.MathSciNetzbMATHGoogle Scholar
  77. [77]
    Chen C B, Moreno Maza M, Xia B C, et al., Computing cylindrical algebraic decomposition via triangular decomposition, ISSAC 2009 — Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2009, 95–102.Google Scholar
  78. [78]
    Wang D M, Elimination Methods, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 2001.Google Scholar
  79. [79]
    Mou C Q and Bai Y, On the chordality of polynomial sets in triangular decomposition in topdown style, ISSAC’18 — Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2018, 287–294.Google Scholar
  80. [80]
    Chen C B and Moreno Maza M, Algorithms for computing triangular decompositions of polynomial systems, ISSAC 2011 — Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2011, 83–90.Google Scholar
  81. [81]
    Wang D M, On the connection between ritt characteristic sets and buchberger-gröbner bases, Math. Comput. Sci., 2016, 10(4): 479–492.MathSciNetzbMATHGoogle Scholar
  82. [82]
    Chai F J, Gao X S, and Yuan C M, A characteristic set method for solving Boolean equations and applications in cryptanalysis of stream ciphers, J. Syst. Sci. Complex., 2008, 21(2): 191–208.MathSciNetzbMATHGoogle Scholar
  83. [83]
    Li X L, Mou C Q, and Wang D M, Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case, Comput. Math. Appl., 2010, 60(11): 2983–2997.MathSciNetzbMATHGoogle Scholar
  84. [84]
    Mou C Q, Wang D M, and Li X L, Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case, Theoret. Comput. Sci., 2013, 468: 102–113.MathSciNetzbMATHGoogle Scholar
  85. [85]
    Gao X S and Huang Z Y, Characteristic set algorithms for equation solving in finite fields, J. Symbolic Comput., 2012, 47(6): 655–679.MathSciNetzbMATHGoogle Scholar
  86. [86]
    Li Z M and Wang D M, Coherent, regular and simple systems in zero decompositions of partial differential systems, Sys. Sci. Math. Sci., 1999, 12: 43–60.zbMATHGoogle Scholar
  87. [87]
    Hubert E, Notes on triangular sets and triangulation-decomposition algorithms. II. Differential systems, Symbolic and Numerical Scientific Computation (Hagenberg, 2001), volume 2630 of Lecture Notes in Comput. Sci., Springer, Berlin, 2003, 40–87.Google Scholar
  88. [88]
    Boulier F, Lemaire F, and Moreno Maza M, Computing differential characteristic sets by change of ordering, J. Symbolic Comput., 2010, 45(1): 124–149.MathSciNetzbMATHGoogle Scholar
  89. [89]
    Rosenfeld A, Specializations in differential algebra, Trans. Amer. Math. Soc., 1959, 90: 394–407.MathSciNetzbMATHGoogle Scholar
  90. [90]
    Gao X S, Huang Z, and Yuan C M, Binomial difference ideals, J. Symbolic Comput., 2017, 80(3): 665–706.MathSciNetGoogle Scholar
  91. [91]
    Mishra B, Algorithmic Algebra, Springer, Boston, MA, 2001.zbMATHGoogle Scholar
  92. [92]
    Bentsen I, The existence of solutions of abstract partial difference polynomials, Trans. Amer. Math. Soc., 1971, 158: 373–397.MathSciNetzbMATHGoogle Scholar
  93. [93]
    Freitag J, Bertini theorems for differential algebraic geometry, ArXiv: 1211.0972v3, 2015.Google Scholar
  94. [94]
    Li W and Li Y H, Computation of differential Chow forms for ordinary prime differential ideals, Adv. in Appl. Math., 2016, 72: 77–112.MathSciNetzbMATHGoogle Scholar
  95. [95]
    Carrà Ferro G, A resultant theory for ordinary algebraic differential equations, Applied algebra, algebraic algorithms and error-correcting codes (Toulouse, 1997), volume 1255 of Lecture Notes in Comput. Sci., Springer, Berlin, 1997, 55–65.Google Scholar
  96. [96]
    Yang L, Zeng Z B, and Zhang W N, Differential elimination with Dixon resultants, Appl. Math. Comput., 2012, 218(21): 10679–10690.MathSciNetzbMATHGoogle Scholar
  97. [97]
    Gao X S, Huang Z, Wang J, et al., Toric difference variety, Journal of Systems Science & Complexity, 2017, 30(1): 173–195.MathSciNetzbMATHGoogle Scholar
  98. [98]
    Golubitsky O D, Kondrat’eva M V, and Ovchinnikov A I, On the generalized Ritt problem as a computational problem, Fundam. Prikl. Mat., 2008, 14(4): 109–120.Google 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.University of Chinese Academy of SciencesBeijingChina

Personalised recommendations