Skip to main content
SpringerLink
Log in

Sparse bivariate polynomial factorization

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

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.

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. Abu Salem F. Factorisation Algorithms for Univariate and Bivariate Polynomials over Finite Fields. PhD thesis. Oxford: Oxford University Computing Laboratory, 2004

    Google Scholar 

  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: 311–341

    Article  MATH  MathSciNet  Google Scholar 

  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, 4–11

    Google Scholar 

  4. Avendaño M, Krick T, Sombra M. Factoring bivariate sparse (lacunary) polynomials. J Complexity, 2007, 23: 193–216

    MATH  MathSciNet  Google Scholar 

  5. Belabas K, van Hoeij M, Klüners J, et al. Factoring polynomials over global fields. J Théorie Nombres Bordeaux, 21: 15–39, 2009

    MATH  Google Scholar 

  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, 1998

    Google Scholar 

  7. Berthomieu J, Lecerf G. Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations. Math Comput, 2012, 81: 1799–1821

    MATH  MathSciNet  Google Scholar 

  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, 42–49

    Google Scholar 

  9. Bürgisser P, Clausen M, Shokrollahi M A. Algebraic Complexity Theory. New York: Springer, 1997

    MATH  Google Scholar 

  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, 141–148

    Google Scholar 

  11. Chéze G, Lecerf G. Lifting and recombination techniques for absolute factorization. J Complexity, 2007, 23: 380–420

    MATH  MathSciNet  Google Scholar 

  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, 2007

    Google Scholar 

  13. Cucker F, Koiran P, Smale S. A polynomial time algorithm for diophantine equations in one variable. J Symbolic Comput, 1999, 27: 21–29

    MATH  MathSciNet  Google Scholar 

  14. Davenport J. Factorisation of sparse polynomials. In: van Hulzen J, ed. Computer Algebra, vol. 162. Lecture Notes in Computer Science. Berlin: Springer, 1983, 214–224

    Google Scholar 

  15. Gao S. Absolute irreducibility of polynomials via Newton polytopes. J Algebra, 2001, 237: 501–520

    MATH  MathSciNet  Google Scholar 

  16. Gao S. Factoring multivariate polynomials via partial differential equations. Math Comput, 2003, 72: 801–822

    MATH  Google Scholar 

  17. Von zur Gathen J. Factoring sparse multivariate polynomials. In: 24th Annual Symposium on Foundations of Computer Science. Tucson: IEEE, 1983, 172–179

    Google Scholar 

  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, 1–2

    Google Scholar 

  19. Von zur Gathen J, Gerhard J. Modern Computer Algebra. London: Cambridge University Press, 1999

    MATH  Google Scholar 

  20. Von zur Gathen J, Kaltofen E. Factoring sparse multivariate polynomials. J Comput Syst Sci, 1985, 31: 265–287

    MATH  Google Scholar 

  21. Geddes K O, Czapor S R, Labahn G. Algorithms for Computer Algebra. Boston: Kluwer Academic Publishers, 1992

    MATH  Google Scholar 

  22. Golub G H, van Loan C. Matrix Computations, 3rd ed. London: The John Hopkins University Press, 1996

    MATH  Google Scholar 

  23. Van Hoeij M. Factoring polynomials and the knapsack problem. J Number Theory, 2002, 95: 167–189

    MATH  MathSciNet  Google Scholar 

  24. Hoppen C, Rodrigues V M, Trevisan V. A note on Gao’s algorithm for polynomial factorization. Theo Comput Sci, 2011, 412: 1508–1522

    MATH  MathSciNet  Google Scholar 

  25. Inaba D. Factorization of multivariate polynomials by extended Hensel construction. ACM SIGSAM Bulletin, 2005, 39: 2–14

    MathSciNet  Google Scholar 

  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, 103–109

    Google Scholar 

  27. Iwami M. Analytic factorization of the multivariate polynomial. In: Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing. Passau, 2003, 213–226

    Google Scholar 

  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, 269–281

    Google Scholar 

  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, 261–266

    Google Scholar 

  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, 57–64

    Google Scholar 

  31. Kaltofen E. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J Comput, 1985, 14: 469–489

    MATH  MathSciNet  Google Scholar 

  32. Kaltofen E. Sparse Hensel lifting. In: Caviness B, ed. EUROCAL’ 85. Lecture Notes in Computer Science, vol. 204. New York: Springer, 1985, 4–17

    Google Scholar 

  33. Kaltofen E. Polynomial factorization 1982–1986. In: Computers and Mathematics. Lecture Notes in Pure and Applied Mathematics, vol. 125. New York: Springer, 1990, 285–309

    Google Scholar 

  34. Kaltofen E. Polynomial factorization 1987–1991. In: Simon I, ed. LATIN’ 92, Lecture Notes in Computer Science, vol. 583. New York: Springer, 1992, 294–313

    Google Scholar 

  35. Kaltofen E. Polynomial factorization: A success story. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation. Philadelphia: ACM, 2003, 3–4

    Google Scholar 

  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, 208–215

    Google Scholar 

  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, 162–168

    Google Scholar 

  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, 382–392

    Google Scholar 

  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, 283–291

    Google Scholar 

  40. Lecerf G. Sharp precision in Hensel lifting for bivariate polynomial factorization. Math Comput, 2006, 75: 921–934

    MATH  MathSciNet  Google Scholar 

  41. Lecerf G. Improved dense multivariate polynomial factorization algorithms. J Symbolic Comput, 2007, 42: 477–494

    MATH  MathSciNet  Google Scholar 

  42. Lecerf G. Fast separable factorization and applications. Appl Algebra Engrg Comm Comput, 2008, 19: 135–160

    MATH  MathSciNet  Google Scholar 

  43. Lecerf G. New recombination algorithms for bivariate polynomial factorization based on Hensel lifting. Appl Algebra Engrg Comm Comput, 2010, 21: 151–176

    MATH  MathSciNet  Google Scholar 

  44. Lenstra A K, Lenstra H W, Lovász L. Factoring polynomials with rational coefficients. Math Ann, 1982, 261: 515–534

    MATH  MathSciNet  Google Scholar 

  45. Lenstra H W. Finding small degree factors of lacunary polynomials. Number Theory, 1999, 1: 267–276

    MathSciNet  Google Scholar 

  46. Lenstra H W. On the factorization of lacunary polynomials. Number Theory, 1999, 1: 277–291

    MathSciNet  Google Scholar 

  47. Musser D R. Multivariate polynomial factorization. J ACM, 1975, 22: 291–308

    MATH  MathSciNet  Google Scholar 

  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, 2007

    Google Scholar 

  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: 9–17

    MATH  Google Scholar 

  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, 159–168

    Google Scholar 

  51. Sasaki T, Kako K. Solving multivariate algebraic equation by Hensel construction. Japan J Indust Appl Math, 1999, 16: 257–285

    MATH  MathSciNet  Google Scholar 

  52. Sasaki T, Saito T, Hilano T. Analysis of approximate factorization algorithm I. Japan J Indust Appl Math, 1992, 9: 351–368

    MATH  MathSciNet  Google Scholar 

  53. Sasaki T, Sasaki M. A unified method for multivariate polynomial factorizations. Japan J Indust Appl Math, 1993, 10: 21–39

    MATH  MathSciNet  Google Scholar 

  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: 357–375

    MATH  MathSciNet  Google Scholar 

  55. Wang P S. Preserving sparseness in multivariate polynominal factorization. In: Proceedings of 1977 MACSYMA Users’ Conference (NASA). Boston: MIT, 1977, 55–64

    Google Scholar 

  56. Wang P S. An improved multivariate polynomial factoring algorithm. Math Comput, 1978, 32: 1215–1231

    MATH  Google Scholar 

  57. Wang P S, Rothschild L P. Factoring multivariate polynomials over the integers. Math Comput, 1975, 29: 935–950

    MATH  MathSciNet  Google Scholar 

  58. Weimann M. A lifting and recombination algorithm for rational factorization of sparse polynomials. J Complexity, 2010, 26: 608–628

    MATH  MathSciNet  Google Scholar 

  59. Weimann M. Factoring bivariate polynomials using adjoints. J Symbolic Comput, 2013, 58: 77–98

    MATH  MathSciNet  Google Scholar 

  60. Wu W, Chen J, Feng Y. An efficient algorithm to factorize sparse bivariate polynomials over the rationals. ACM Commun Comput Algebra, 2012, 46: 125–126

    Google Scholar 

  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, 216–226

    Google Scholar 

  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, 68–72

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, 400714, China

    WenYuan Wu, JingWei Chen & Yong Feng

Authors
  1. WenYuan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. JingWei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JingWei Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, W., Chen, J. & Feng, Y. Sparse bivariate polynomial factorization. Sci. China Math. 57, 2123–2142 (2014). https://doi.org/10.1007/s11425-014-4850-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4850-y

Keywords

MSC(2010)

Search

Navigation