Polynomial Factorization 1987-1991

Algorithms invented in the past 25 years make it possible on a computer to efficiently factor a polynomial in one, several, or many variables with coefficients from a certain field, such as a finite field or the rational, real, or complex numbers. I have surveyed work up to 1986 in the papers (Kaltofen 1982 and 1990a). This article discusses important developments of the past five years; I also take a fresh perspective of some older results. Although a conscientious effort has been made to cover (at least by citation) the significant contributions of that period, omissions are likely, which I ask to be kindly brought to my attention. Three parameters partition the factorization problem: first, the mathematical nature and computational representation of the coefficient domains of the input polynomial, second, that of the irreducible factors, and, third, the representation of the input polynomial and the sought irreducible factors, which depends not only on the degree and number of variables but also on properties such as sparsity. Say, for instance, that a bivariate polynomial with rational coefficients is to be factored into irreducible polynomials with real coefficients. The input polynomial as well as the factors may be represented by lists of monomials, that is terms and their corresponding non-zero coefficients. For the rational input the coefficients can be just fractions of two long integers, but the representation of the real coefficients for the factors is less standardized. One choice represents a real algebraic number by its rational minimum polynomial and an isolating interval with rational boundaries (Collins 1975), while another uses a rational linear relation of powers of a complex algebraic number that is universal for all coefficients of a single factor (Kaltofen 1990b). The organization of this survey is governed by these distinguishing problem specifications. We first discuss the “classical univariate problems” of factoring a polynomial