Sparse hensel lifting

  • Erich Kaltofen
Algebraic Algorithms I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 204)


A new algorithm is introduced which computes the multivariate leading coefficients of polynomial factors from their univariate images. This algorithm is incorporated into a sparse Hensel lifting scheme and only requires the factorization of a single univariate image. The algorithm also provides the content of the input polynomial in the main variable as a by-product. We show how we can take advantage of this property when computing the GCD of multivariate polynomials by sparse Hensel lifting.


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  1. [1]
    Bareiss, E.H.: Sylvester's identity and multistep integers preserving Gaussian elimination. Math. Comp. 22, 565–578 (1965).Google Scholar
  2. [2]
    Brown, W.S.: On Euclid's algorithm and the computation of polynomial greatest common divisors. J. ACM 18, 478–504 (1971).Google Scholar
  3. [3]
    Buchberger, B., Loos, R.: Algebraic Simplification. Computing, Supplement 4, 11–43 (1982).Google Scholar
  4. [4]
    Czapor, S.R., Geddes, K.O.: A comparison of algorithms for the symbolic computation of Padé approximants. Proc. EUROSAM 1984, Springer Lec. Notes Comp. Sci. 174, 248–259.Google Scholar
  5. [5]
    Epstein, H.I.: Using basis computation to determine pseudo-multiplicative independence. Proc. 1976 ACM Symp. Symbolic Algebraic Comp., 229–237.Google Scholar
  6. [6]
    von zur Gathen, J.: Irreducibility of multivariate polynomials. J. Comp. System Sci., to appear.Google Scholar
  7. [7]
    von zur Gathen, J.: Factoring sparse multivariate polynomials. Proc. 24th IEEE Symp. Foundations Comp. Sci., 172–179 (1983).Google Scholar
  8. [8]
    Kaltofen, E.: Effective Hilbert Irreducibility. Proc. EUROSAM 1984, Springer Lec. Notes Comp. Sci. 174, 277–284.Google Scholar
  9. [9]
    Kaltofen, E.: On a theorem by R. Dedekind. Manuscript 1984.Google Scholar
  10. [10]
    Kaltofen, E.: Computing with polynomials given by straight-line programs I; Greatest Common Divisors. Proc. 17th ACM Symp. Theory Comp., to appear (1985).Google Scholar
  11. [11]
    Kaltofen, E.: Computing with polynomials given by straight-line programs II; Factorization. In preparation.Google Scholar
  12. [12]
    Moses, J., Yun, D.Y.Y.: The EZ-GCD algorithm. Proc 1973 ACM National Conf., 159–166.Google Scholar
  13. [13]
    Schwarz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27, 701–717 (1980).CrossRefGoogle Scholar
  14. [14]
    Viry, G.: Factorisation des polynomes a plusieurs variables. R.A.I.R.O. Informatique Théoretique 17, 209–223 (1980).Google Scholar
  15. [15]
    Wang, P.S.: An improved multivariate polynomial factorization algorithm. Math. Comp. 32, 1215–1231 (1978).Google Scholar
  16. [16]
    Wang, P.S.: The EEZ-GCD algorithm. SIGSAM Bulletin 14-2, 50–60 (May 1980).CrossRefGoogle Scholar
  17. [17]
    Wang, P.S.: Early detection of true factors in univariate polynomial factorization. Proc. EUROCAL 1983, Springer Lec. Notes Comp. Sci. 162, 225–235.Google Scholar
  18. [18]
    Yun, D.Y.Y.: The Hensel lemma in algebraic manipulation. Ph.D. dissertation, MIT 1974. Reprint: Garland Publ. Co., New York 1980.Google Scholar
  19. [19]
    Zippel, R.E.: Newton's iteration and the sparse Hensel algorithm. Proc. 1981 ACM Symp. Symbolic Algebraic Comp., 68–72.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Erich Kaltofen
    • 1
  1. 1.Rensselaer Polytechnic InstituteDepartment of Computer ScienceTroy

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