Newton’s Iteration and the Hensel Construction

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