Science China Mathematics

, Volume 57, Issue 10, pp 2123–2142 | Cite as

Sparse bivariate polynomial factorization

  • WenYuan Wu
  • JingWei Chen
  • Yong Feng


Motivated by Sasaki’s work on the extended Hensel construction for solving multivariate algebraic equations, we present a generalized Hensel lifting, which takes advantage of sparsity, for factoring bivariate polynomial over the rational number field. Another feature of the factorization algorithm presented in this article is a new recombination method, which can solve the extraneous factor problem before lifting based on numerical linear algebra. Both theoretical analysis and experimental data show that the algorithm is efficient, especially for sparse bivariate polynomials.


polynomial factorization sparse polynomial generalized Hensel lifting 


12Y05 68W30 11Y16 12D05 13P05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abu Salem F. Factorisation Algorithms for Univariate and Bivariate Polynomials over Finite Fields. PhD thesis. Oxford: Oxford University Computing Laboratory, 2004Google Scholar
  2. 2.
    Abu Salem F. An efficient sparse adaptation of the polytope method over \(\mathbb{F}_p\) and a record-high binary bivariate factorisation. J Symbolic Comput, 2008, 43: 311–341CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Abu Salem F, Gao S, Lauder A G B. Factoring polynomials via polytopes. In: Proceedings of the 2004 international symposium on Symbolic and algebraic computation. Santander: ACM, 2004, 4–11Google Scholar
  4. 4.
    Avendaño M, Krick T, Sombra M. Factoring bivariate sparse (lacunary) polynomials. J Complexity, 2007, 23: 193–216MATHMathSciNetGoogle Scholar
  5. 5.
    Belabas K, van Hoeij M, Klüners J, et al. Factoring polynomials over global fields. J Théorie Nombres Bordeaux, 21: 15–39, 2009MATHGoogle Scholar
  6. 6.
    Bernardin L. On bivariate Hensel and its parallelization. In: Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation. New York: ACM, 96–100, 1998Google Scholar
  7. 7.
    Berthomieu J, Lecerf G. Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations. Math Comput, 2012, 81: 1799–1821MATHMathSciNetGoogle Scholar
  8. 8.
    Bostan A, Lecerf G, Salvy B, et al. Complexity issues in bivariate polynomial factorization. In: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation. Santander: ACM, 2004, 42–49Google Scholar
  9. 9.
    Bürgisser P, Clausen M, Shokrollahi M A. Algebraic Complexity Theory. New York: Springer, 1997MATHGoogle Scholar
  10. 10.
    Chattopadhyay A, Grenet B, Koiran P, et al. Factoring bivariate lacunary polynomials without heights. In: Proceedings of the 2013 International Symposium on Symbolic and Algebraic Computation. Boston: ACM, 2013, 141–148Google Scholar
  11. 11.
    Chéze G, Lecerf G. Lifting and recombination techniques for absolute factorization. J Complexity, 2007, 23: 380–420MATHMathSciNetGoogle Scholar
  12. 12.
    Cox D, Little J, O’Shea D. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed. New York: Springer, 2007Google Scholar
  13. 13.
    Cucker F, Koiran P, Smale S. A polynomial time algorithm for diophantine equations in one variable. J Symbolic Comput, 1999, 27: 21–29MATHMathSciNetGoogle Scholar
  14. 14.
    Davenport J. Factorisation of sparse polynomials. In: van Hulzen J, ed. Computer Algebra, vol. 162. Lecture Notes in Computer Science. Berlin: Springer, 1983, 214–224Google Scholar
  15. 15.
    Gao S. Absolute irreducibility of polynomials via Newton polytopes. J Algebra, 2001, 237: 501–520MATHMathSciNetGoogle Scholar
  16. 16.
    Gao S. Factoring multivariate polynomials via partial differential equations. Math Comput, 2003, 72: 801–822MATHGoogle Scholar
  17. 17.
    Von zur Gathen J. Factoring sparse multivariate polynomials. In: 24th Annual Symposium on Foundations of Computer Science. Tucson: IEEE, 1983, 172–179Google Scholar
  18. 18.
    Von zur Gathen J. Who was who in polynomial factorization. In: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation. Genova: ACM, 2006, 1–2Google Scholar
  19. 19.
    Von zur Gathen J, Gerhard J. Modern Computer Algebra. London: Cambridge University Press, 1999MATHGoogle Scholar
  20. 20.
    Von zur Gathen J, Kaltofen E. Factoring sparse multivariate polynomials. J Comput Syst Sci, 1985, 31: 265–287MATHGoogle Scholar
  21. 21.
    Geddes K O, Czapor S R, Labahn G. Algorithms for Computer Algebra. Boston: Kluwer Academic Publishers, 1992MATHGoogle Scholar
  22. 22.
    Golub G H, van Loan C. Matrix Computations, 3rd ed. London: The John Hopkins University Press, 1996MATHGoogle Scholar
  23. 23.
    Van Hoeij M. Factoring polynomials and the knapsack problem. J Number Theory, 2002, 95: 167–189MATHMathSciNetGoogle Scholar
  24. 24.
    Hoppen C, Rodrigues V M, Trevisan V. A note on Gao’s algorithm for polynomial factorization. Theo Comput Sci, 2011, 412: 1508–1522MATHMathSciNetGoogle Scholar
  25. 25.
    Inaba D. Factorization of multivariate polynomials by extended Hensel construction. ACM SIGSAM Bulletin, 2005, 39: 2–14MathSciNetGoogle Scholar
  26. 26.
    Inaba D, Sasaki T. A numerical study of extended Hensel series. In: Proceedings of the 2007 International Workshop on Symbolic-numeric Computation. London-Canada: ACM, 2007, 103–109Google Scholar
  27. 27.
    Iwami M. Analytic factorization of the multivariate polynomial. In: Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing. Passau, 2003, 213–226Google Scholar
  28. 28.
    Iwami M. Extension of expansion base algorithm to multivariate analytic factorization. In: Proceedings of the 7th International Workshop on Computer Algebra in Scientific Computing. Petersburg, 2004, 269–281Google Scholar
  29. 29.
    Kaltofen E. A polynomial reduction from multivariate to bivariate integral polynomial factorization. In: Proceedings of the 14th Annual ACM Symposium on Theory of Computing. New York: ACM, 1982, 261–266Google Scholar
  30. 30.
    Kaltofen E. A polynomial-time reduction from bivariate to univariate integral polynomial factorization. In: 23rd Annual Symposium on Foundations of Computer Science. Chicago: IEEE, 1982, 57–64Google Scholar
  31. 31.
    Kaltofen E. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J Comput, 1985, 14: 469–489MATHMathSciNetGoogle Scholar
  32. 32.
    Kaltofen E. Sparse Hensel lifting. In: Caviness B, ed. EUROCAL’ 85. Lecture Notes in Computer Science, vol. 204. New York: Springer, 1985, 4–17Google Scholar
  33. 33.
    Kaltofen E. Polynomial factorization 1982–1986. In: Computers and Mathematics. Lecture Notes in Pure and Applied Mathematics, vol. 125. New York: Springer, 1990, 285–309Google Scholar
  34. 34.
    Kaltofen E. Polynomial factorization 1987–1991. In: Simon I, ed. LATIN’ 92, Lecture Notes in Computer Science, vol. 583. New York: Springer, 1992, 294–313Google Scholar
  35. 35.
    Kaltofen E. Polynomial factorization: A success story. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation. Philadelphia: ACM, 2003, 3–4Google Scholar
  36. 36.
    Kaltofen E, Koiran P. On the complexity of factoring bivariate supersparse (lacunary) polynomials. In: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation. Beijing: ACM, 2005, 208–215Google Scholar
  37. 37.
    Kaltofen E, Koiran P. Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields. In: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation. Genoa: ACM, 2006, 162–168Google Scholar
  38. 38.
    Kaltofen E, Lecerf G. Factorization of multivairate polynomials. In: Mullen G L, Panario D, eds. Handbook of Finite Fields. Boca Raton: CRC Press, 2013, 382–392Google Scholar
  39. 39.
    Klüners J. The van Hoeij algorithm for factoring polynomials. In: Nguyen P Q, Vallée B, eds. The LLL Algorithm: Survey and Applications. New York: Springer, 2010, 283–291Google Scholar
  40. 40.
    Lecerf G. Sharp precision in Hensel lifting for bivariate polynomial factorization. Math Comput, 2006, 75: 921–934MATHMathSciNetGoogle Scholar
  41. 41.
    Lecerf G. Improved dense multivariate polynomial factorization algorithms. J Symbolic Comput, 2007, 42: 477–494MATHMathSciNetGoogle Scholar
  42. 42.
    Lecerf G. Fast separable factorization and applications. Appl Algebra Engrg Comm Comput, 2008, 19: 135–160MATHMathSciNetGoogle Scholar
  43. 43.
    Lecerf G. New recombination algorithms for bivariate polynomial factorization based on Hensel lifting. Appl Algebra Engrg Comm Comput, 2010, 21: 151–176MATHMathSciNetGoogle Scholar
  44. 44.
    Lenstra A K, Lenstra H W, Lovász L. Factoring polynomials with rational coefficients. Math Ann, 1982, 261: 515–534MATHMathSciNetGoogle Scholar
  45. 45.
    Lenstra H W. Finding small degree factors of lacunary polynomials. Number Theory, 1999, 1: 267–276MathSciNetGoogle Scholar
  46. 46.
    Lenstra H W. On the factorization of lacunary polynomials. Number Theory, 1999, 1: 277–291MathSciNetGoogle Scholar
  47. 47.
    Musser D R. Multivariate polynomial factorization. J ACM, 1975, 22: 291–308MATHMathSciNetGoogle Scholar
  48. 48.
    Press W H, Teukolsky S A, Vetterling W T, et al. Numerical Recipes: The Art of Scientific Computing, 3rd ed. New York: Cambridge University Press, 2007Google Scholar
  49. 49.
    Sasaki T, Inaba D. Hensel construction of f(x, u 1, …, u l), l ⩾ 2, at a singular point and its applications. ACM SIGSAM Bulletin, 2000, 34: 9–17MATHGoogle Scholar
  50. 50.
    Sasaki T, Inaba D. Convergence and many-valuedness of Hensel series near the expansion point. In: Proceedings of the 2009 Conference on Symbolic Numeric Computation. Kyoto: ACM, 2009, 159–168Google Scholar
  51. 51.
    Sasaki T, Kako K. Solving multivariate algebraic equation by Hensel construction. Japan J Indust Appl Math, 1999, 16: 257–285MATHMathSciNetGoogle Scholar
  52. 52.
    Sasaki T, Saito T, Hilano T. Analysis of approximate factorization algorithm I. Japan J Indust Appl Math, 1992, 9: 351–368MATHMathSciNetGoogle Scholar
  53. 53.
    Sasaki T, Sasaki M. A unified method for multivariate polynomial factorizations. Japan J Indust Appl Math, 1993, 10: 21–39MATHMathSciNetGoogle Scholar
  54. 54.
    Sasaki T, Suzuki M, Kolář M, et al. Approximate factorization of multivariate polynomials and absolute irreducibility testing. Japan J Indust Appl Math, 1991, 8: 357–375MATHMathSciNetGoogle Scholar
  55. 55.
    Wang P S. Preserving sparseness in multivariate polynominal factorization. In: Proceedings of 1977 MACSYMA Users’ Conference (NASA). Boston: MIT, 1977, 55–64Google Scholar
  56. 56.
    Wang P S. An improved multivariate polynomial factoring algorithm. Math Comput, 1978, 32: 1215–1231MATHGoogle Scholar
  57. 57.
    Wang P S, Rothschild L P. Factoring multivariate polynomials over the integers. Math Comput, 1975, 29: 935–950MATHMathSciNetGoogle Scholar
  58. 58.
    Weimann M. A lifting and recombination algorithm for rational factorization of sparse polynomials. J Complexity, 2010, 26: 608–628MATHMathSciNetGoogle Scholar
  59. 59.
    Weimann M. Factoring bivariate polynomials using adjoints. J Symbolic Comput, 2013, 58: 77–98MATHMathSciNetGoogle Scholar
  60. 60.
    Wu W, Chen J, Feng Y. An efficient algorithm to factorize sparse bivariate polynomials over the rationals. ACM Commun Comput Algebra, 2012, 46: 125–126Google Scholar
  61. 61.
    Zippel R. Probabilistic algorithms for sparse polynomials. In: Ng E, ed. Symbolic and Algebraic Computation. Lecture Notes in Computer Science, vol. 72. London: Springer, 1979, 216–226Google Scholar
  62. 62.
    Zippel R. Newton’s iteration and the sparse Hensel algorithm (extended abstract). In: Proceedings of the 4th ACM Symposium on Symbolic and Algebraic Computation. Snowbird: ACM, 1981, 68–72Google Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent TechnologyChinese Academy of SciencesChongqingChina

Personalised recommendations