Using Sparse Interpolation in Hensel Lifting

  • Michael MonaganEmail author
  • Baris Tuncer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9890)


The standard approach to factor a multivariate polynomial in \(\mathbb {Z}[x_{1},x_{2},\dots ,x_{n}]\) is to factor a univariate image in \(\mathbb {Z}[x_{1}]\) then lift the factors of the image one variable at a time using Hensel lifting to recover the multivariate factors. At each step one must solve a multivariate polynomial Diophantine equation. For polynomials in many variables with many terms we find that solving these multivariate Diophantine equations dominates the factorization time. In this paper we explore the use of sparse interpolation methods, originally introduced by Zippel, to speed this up. We present experimental results in Maple showing that we are able to dramatically speed this up and thereby achieve a good improvement for multivariate polynomial factorization.


Ideal Type Evaluation Cost Toeplitz Matrix Taylor Coefficient Monic Factor 
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  1. 1.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms, 3rd edn. Springer, Heidleberg (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Geddes, K.O., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer, Boston (1992)CrossRefzbMATHGoogle Scholar
  3. 3.
    Kaltofen, E.: Sparse Hensel Lifting. In: Caviness, B.F. (ed.) EUROCAL 1985. LNCS, vol. 204, pp. 4–17. Springer, Heidelberg (1985)Google Scholar
  4. 4.
    Lee, M.M.: Factorization of multivariate polynomials. Ph.D. Thesis (2013)Google Scholar
  5. 5.
    Miola, A., Yun, D.Y.Y.: Computational aspects of Hensel-type univariate polynomial greatest common divisor algorithms. In: Proceedings of EUROSAM 1974, pp. 46–54. ACM Press (1974)Google Scholar
  6. 6.
    Monagan, M.B., Tuncer, B.: Some results on counting roots of polynomials and the Sylvester resultant. In: Proceedings of FPSAC 2016. DMTCS (to appear 2016)Google Scholar
  7. 7.
    Monagan, M.B., Pearce, R.: POLY: a new polynomial data structure for Maple 17. In: Feng, R., Lee, W.-S., Sato, Y. (eds.) Computer Mathematics, pp. 325–348. Springer, Heidelberg (2014)Google Scholar
  8. 8.
    Schwartz, J.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27, 701–717 (1980). ACM PressMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wang, P.S.: An improved multivariate polynomial factoring algorithm. Math. Comput. 32, 1215–1231 (1978). AMSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wang, P.S., Rothschild, L.P.: Factoring multivariate polynomials over the integers. Math. Comput. 29(131), 935–950 (1975). AMSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yun, D.Y.Y.: The Hensel Lemma in algebraic manipulation. Ph.D. Thesis (1974)Google Scholar
  12. 12.
    Zippel, R.: Interpolating polynomials from their values. J. Symbolic Comput. 9, 375–403 (1990). Academic PressMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zippel, R.E.: Probabilistic algorithms for sparse polynomials. In: Proceedings of EUROSAM 1979. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979)Google Scholar
  14. 14.
    Zippel, R.E.: Newton’s iteration and the sparse Hensel algorithm. In: Proceedings of SYMSAC 1981, pp. 68–72. ACM Press (1981)Google Scholar

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© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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