Journal of Systems Science and Complexity

, Volume 28, Issue 1, pp 243–260 | Cite as

Exact bivariate polynomial factorization over ℚ by approximation of roots

  • Yong Feng
  • Wenyuan Wu
  • Jingzhong Zhang
  • Jingwei Chen


Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not support symbolic computation directly. Hence, it leads to difficulties in applying factorization in engineering fields. In this paper, the authors present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients. The proposed method can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library. In addition, the numerical computation part often only requires double precision and is easily parallelizable.


Factorization of multivariate polynomials interpolation methods minimal polynomial numerical continuation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Zassenhaus H, On hensel factorization, I, Journal of Number Theory, 1969, 1(3): 291–311.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    Zassenhaus H, A remark on the hensel factorization method, Mathematics of Computation, 1978, 32(141): 287–292.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    von zur Gathen J, Irreducibility of multivariate polynomials, Journal of Computer and System Sciences, 1985, 31(2): 225–264.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    von zur Gathen J and Gerhard J, Modern Computer Algebra, 2nd edition, Cambridge University Press, London, 2003.zbMATHGoogle Scholar
  5. [5]
    Lecerf G, Sharp precision in Hensel lifting for bivariate polynomial factorization, Mathematics of Computation, 2006, 75(254): 921–934.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Lecerf G, Improved dense multivariate polynomial factorization algorithms, Journal of Symbolic Computation, 2007, 42(4): 477–494.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Lecerf G, Fast separable factorization and applications, Applicable Algebra in Engineering, Communication and Computing, 2008, 19(2): 135–160.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Lecerf G, New recombination algorithms for bivariate polynomial factorization based on Hensel lifting, Applicable Algebra in Engineering, Communication and Computing, 2010, 21(2): 151–176.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Berthomieu J and Lecerf G, Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations, Mathematics of Computation, 2012, 81(279): 1799–1821.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Weimann M, Factoring bivariate polynomials using adjoints, Journal of Symbolic Computation, 2013, 58: 77–98.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    Abu Salem F, An efficient sparse adaptation of the polytope method over \(\mathbb{F}_p \) and a record-high binary bivariate factorisation, Journal of Symbolic Computation, 2008, 43(5): 311–341.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Weimann M, A lifting and recombination algorithm for rational factorization of sparse polynomials, Journal of Complexity, 2010, 26(6): 608–628.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Chattopadhyay A, Grenet B, Koiran P, Portier N, and Strozecki Y, Factoring bivariate lacunary polynomials without heights, in Proceedings of the 2013 international symposium on Symbolic and Algebraic Computation, Boston, USA, 2013, 141–148.Google Scholar
  14. [14]
    Wu W, Chen J, and Feng Y, Sparse bivariate polynomial factorization, Science China Mathematics, 2014, 57, doi: 10.1007/s11425-014-4850-y.Google Scholar
  15. [15]
    Kaltofen E and Yagati L, Improved sparse multivariate polynomial interpolation algorithms, in Gianni P, editor, Symbolic and Algebraic Computation, Lecture Notes in Computer Science, Springer, 1989, 358: 467–474.CrossRefMathSciNetGoogle Scholar
  16. [16]
    Mou-Yan Z and Unbehauen R, Approximate factorization of multivariable polynomials, Signal Processing, 1988, 14(2): 141–152.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Huang Y, Wu W, Stetter H J, and Zhi L, Pseudofactors of multivariate polynomials, in Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, St. Andrews, Scotland, 2000, 161–168.Google Scholar
  18. [18]
    Sasaki T, Suzuki M, Kol′ař M, and Sasaki M, Approximate factorization of multivariate polynomials and absolute irreducibility testing, Japan Journal of Industrial and Applied Mathematics, 1991, 8(3): 357–375.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    Sasaki T, Saito T, and Hilano T, Analysis of approximate factorization algorithm I, Japan Journal of Industrial and Applied Mathematics, 1992, 9(3): 351–368.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    Sasaki T, Approximate multivariate polynomial factorization based on zero-sum relations, in Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, London, Canada, 2001, 284–291.Google Scholar
  21. [21]
    Corless R M, Giesbrecht MW, van Hoeij M, Kotsireas I, and Watt S M, Towards factoring bivariate approximate polynomials, in Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, London, Canada, 2001, 85–92.Google Scholar
  22. [22]
    Sommese A J, Verschelde J, and Wampler C W, Numerical factorization of multivariate complex polynomials, Theoretical Computer Science, 2004, 315(2–3): 651–669.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    Gao S, Kaltofen E, May J P, Yang Z, and Zhi L, Approximate factorization of multivariate polynomials via differential equations, in Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, Santander, Spain, 2004, 167–174.Google Scholar
  24. [24]
    Kaltofen E, May J P, Yang Z, and Zhi L, Approximate factorization of multivariate polynomials using singular value decomposition, Journal of Symbolic Computation, 2008, 43(5): 359–376.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    Gao S, Factoring multivariate polynomials via partial differential equations, Mathematics of Computation, 2003, 72(242): 801–822.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    Rupprecht D, Semi-numerical absolute factorization of polynomials with integer coefficients, Journal of Symbolic Computation, 2004, 37(5): 557–574.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    Ch′eze G and Galligo A, From an approximate to an exact absolute polynomial factorization, Journal of Symbolic Computation, 2006, 41(6): 682–696.CrossRefMathSciNetGoogle Scholar
  28. [28]
    Kannan R, Lenstra A K, and Lov′asz L, Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers, Mathematics of Computation, 1988, 50(181): 235–250.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    Zhang J and Feng Y, Obtaining exact value by approximate computations, Science in China Series A, Mathematics, 2007, 50(9): 1361–1368.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    Qin X, Feng Y, Chen J, and Zhang J, A complete algorithm to find exact minimal polynomial by approximations, International Journal of Computer Mathematics, 2012, 89(17): 2333–2344.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    van der Hulst M-P and Lenstra A K, Factorization of polynomials by transcendental evaluation, in Caviness B F, editor, EUROCAL’ 85, Lecture Notes in Computer Science, 1985, 204: 138–145.CrossRefGoogle Scholar
  32. [32]
    Chen J, Feng Y, Qin X, and Zhang J, Exact polynomial factorization by approximate high degree algebraic numbers, Proceedings of the 2009 Conference on Symbolic Numeric Computation, Kyoto, Japan, 2009, 21–28.Google Scholar
  33. [33]
    Lee T L, Li T Y, and Tsai C H, HOM4PS-2.0, a software package for solving polynomial systems by the polyhedral homotopy continuation method, Computing, 2008, 83(2–3): 109–133.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    Bates D J, Hauenstein J D, Sommese A J, and Wampler C W, Bertini, Software for numerical algebraic geometry, Available at, June 2014.Google Scholar
  35. [35]
    Verschelde J, Algorithm 795, PHCpack, A general-purpose solver for polynomial systems by homotopy continuation, ACM Transactions on Mathematical Software, 1999, 25(2): 251–276.CrossRefzbMATHGoogle Scholar
  36. [36]
    Blum L, Cucker F, Shub M, and Smale S, Complexity and Real Computation, Springer, New York, 1998.CrossRefGoogle Scholar
  37. [37]
    Ilie S, Corless R M, and Reid G, Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time, Numerical Algorithms, 2006, 41(2): 161–171.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    Morgan A and Sommese A J, A homotopy for solving general polynomial systems that respects m-homogeneous structures, Applied Mathematics and Computation, 1987, 24(2): 101–113.CrossRefMathSciNetGoogle Scholar
  39. [39]
    Sommese A J and Wampler C W, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore, 2005.CrossRefzbMATHGoogle Scholar
  40. [40]
    Fried M, On Hilbert’s rrreducibility theorem, Journal of Number Theory, 1974, 6(3): 211–231.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Yong Feng
    • 1
  • Wenyuan Wu
    • 1
  • Jingzhong Zhang
    • 1
  • Jingwei Chen
    • 1
  1. 1.Chongqing Key Lab of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology (CIGIT)Chinese Academy of SciencesChongqingChina

Personalised recommendations