Algorithms for Computer Algebra pp 205-277 | Cite as

# Newton’s Iteration and the Hensel Construction

## Abstract

In this chapter we continue our discussion of techniques for inverting modular and evaluation homomorphisms defined on the domain **Z**[*x* _{1}, . . ., *x* _{v} ]. The particular methods developed in this chapter are based on Newton's iteration for solving a polynomial equation. Unlike the integer and polynomial Chinese remainder algorithms of the preceding chapter, algorithms based on Newton's iteration generally require only one image of the solution in a domain of the form **Z** _{ p }[*x* _{1}] from which to reconstruct the desired solution in the larger domain **Z**[*x* _{1}, . . . , *x* _{v}]. A particularly important case of Newton's iteration to be discussed here is the *Hensel construction*. It will be seen that multivariate polynomial computations (such as GCD computation and factorization) can be performed much more efficiently (in most cases) by methods based on the Hensel construction than by methods based on the Chinese remainder algorithms of the preceding chapter

## Keywords

Iteration Step Computer Algebra Multivariate Polynomial Prime Integer Degree Constraint## Preview

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## References

- 1.M. Lauer, “Computing by Homomorphic Images,” pp. 139–168 in
*Computer Algebra — Symbolic and Algebraic Computation*, ed. B. Buchberger, G.E. Collins and R. Loos, Springer-Verlag (1982).Google Scholar - 2.J.D. Lipson, “Newton’s Method: A Great Algebraic Algorithm,” pp. 260–270 in
*Proc. SYMSAC’ 76*, ed. R.D. Jenks, ACM Press (1976).Google Scholar - 3.R. Loos, “Rational Zeros of Integral Polynomials by p-Adic Expansions,”
*SIAM J. on Computing*, 12 pp. 286–293 (1983).MATHCrossRefMathSciNetGoogle Scholar - 4.M. Mignotte, “Some Useful Bounds.,” pp. 259–263 in
*Computer Algebra — Symbolic and Algebraic Computation*, ed. B. Buchberger, G.E. Collins and R. Loos, Springer-Verlag (1982).Google Scholar - 5.A. Miola and D.Y.Y. Yun, “The Computational Aspects of Hensel-Type Univariate Greatest Common Divisor Algorithms,”
*(Proc. EUROSAM’ 74) ACM SIGSAM Bull.*,**8**(3) pp. 46–54 (1974).CrossRefGoogle Scholar - 6.P.S. Wang, “An Improved Multivariate Polynomial Factoring Algorithm,”
*Math. Comp.*,**32**pp. 1215–1231 (1978).MATHMathSciNetGoogle Scholar - 7.P.S. Wang, “The EEZ-GCD Algorithm,”
*ACM SIGSAM Bull.*,**14**pp. 50–60 (1980).MATHCrossRefGoogle Scholar - 8.D.Y.Y. Yun, “The Hensel Lemma in Algebraic Manipulation,” Ph.D. Thesis, M.I.T. (1974).Google Scholar
- 9.H. Zassenhaus, “Hensel Factorization I,”
*J. Number Theory*,**1**pp. 291–311 (1969).MATHCrossRefMathSciNetGoogle Scholar - 10.R. Zippel, “Newton’s Iteration and the Sparse Hensel Algorithm,” pp. 68–72 in
*Proc. SYMSAC’ 81*, ed. P.S. Wang, ACM Press (1981).Google Scholar