An improved multivariate polynomial factoring algorithm

Abstract

A new algorithm for factoring multivariate polynomials over the integers based on an algorithm by Wang and Rothschild is described. The new algorithm has improved strategies for dealing with the known problems of the original algorithm, namely, the leading coefficient problem, the bad-zero problem and the occurrence of extraneous factors. It has an algorithm for correctly predetermining leading coefficients of the factors. A new and efficient p-adic algorithm named EEZ is described. Basically it is a linearly convergent variable-by-variable parallel construction. The improved algorithm is generally faster and requires less store then the original algorithm. Machine examples with comparative timing are included.

History of finite fields

“…This approach was originally popularized in computer algebra by Zassenhaus in [Zassenhaus 1969] in the context of factoring in Q[y] via the p-adic completion of Q. The adaptation to two and several variables was first pioneered in [Musser 1975;Wang 1978;Wang and Rothschild 1975]. In particular, [Musser 1975] introduced coefficient field abstractions that marked the beginning of generic programming.…”

Computer Algebra Algorithms

“…GCD and factorization algorithms on such multivariate sparse polynomials were studied both theoretically and empirically as early as a decade ago (Wang 1978) [177], and randomization became one of the major ingredients to sparsity preserving operations (Zippel 1979) [188]. Early on Moses (1971b) [125] pointed out, however, that sparsity is not measured in terms of nonzero coefficients alone, as his example…”

On factorization of multivariate polynomials over algebraic number and function fields

“…Abbott in [1], suggests using a trick by Kaltofen in [4] which recursively computes the leading coefficients from their bivariate images using Hensel lifting. Our approach is to modify Wang's ingenious method given in [8] for factoring polynomials over Z. His idea is to first factor the leading coefficient l(x2, .…”