Factorization of Polynomials Given by Straight-Line Programs

An algorithm is developed for the factorization of a multivariate polynomial represented by a straight-line program into its irreducible factors. The algorithm is in random polynomial-time as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and outputs a straight-line program, which with controllably high probability correctly determines the irreducible factors. It also returns the probably correct multiplicities of each distinct factor. If the coefficient field has finite characteristic p and p divides the multiplicities of some irreducible factors our algorithm constructs straight-line programs for the appropriate p-th powers of such factors. Also a probabilistic algorithm is presented that allows to convert a polynomial given by a straightline program into its sparse representation. This conversion algorithm is in random-polynomial time in the previously cited parameters and in an upper bound for the number of non-zero monomials permitted in the sparse output. Together with our factorization algorithm we therefore can probabilistically determine all those sparse irreducible factors of a polynomial given by a straight-line program that have less than a given number of monomials. We show that this result is valid without any restriction on the characteristic of the coefficient field. The first section of this paper also summarizes the history of the polynomial factorization problem, and the last section discusses what questions for this problem remain to be resolved. We hav e also attempted to provide an extensive list of references on the subject, so that this paper can serve as a starting point for someone without previous knowledge in polynomial factorization. 1. The Problem of Factoring Polynomials * This material is based upon work supported by the National Science Foundation under Grant No. DCR-85-04391 and by an IBM Faculty Development Award. The results of §6 were originally announced in the paper “Uniform closure properties of p-computable functions” [32]. This paper appears in Randomness and Computation, edited by S. Micali, vol. 5 of the Advances in Computing Research series, JAI Press Inc., Greenwich, Connecticut, pp. 375-412 (1989).

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