Square-free Factorization Research Articles

An algorithm for avoiding complex numbers in rational function integration

Given a proper rational function A(x)/B(x) where A(x) and B(x) both are in R[x] with gcd(A(x), B(x))= 1, B(x) monic and deg(A(x)) < deg(B(x)), from the Hermite algorithm for rational function integration in [3], we obtain[EQUATION]where S(x) is a rational function which is called the rational part of the integral of A(x)/B(x) in eq. (1), B*(x) is the greatest square-free factor of B(x), and T(x) is in R[x] with deg(T(x)) < deg(B*(x)). The integral of T(x)/B*(x) is called the transcendental part of the integral of A(x)/B(x) in eq. (1).

New Algorithms for Polynomial Square-Free Decomposition over the Integers

Previously-known algorithms for polynomial square-free decomposition rely on greatest common divisor (gcd) computations over the same coefficient domain where the decomposition is to be performed. In particular, gcd of the given polynomial and its first derivative (with respect to some variable) is obtained to begin with. Application of modular homomorphism and p-adic construction (multivariate case) or the Chinese remainder algorithm (univariate case) results in new square-free decomposition algorithms which, generally speaking, take less time than a single gcd between the given polynomial and its first derivative. The key idea is to obtain one or several “correct” homomorphic images of the desired square-free decomposition first. This provides information as to how many different square-free factors there are, their multiplicities and their homomorphic images. Since the multiplicities are known, only the square-free factors need be constructed. Thus, these new algorithms are relatively insensitive to th...