For the general case, we give a random (Monte-Carlo) NC algorithm in these input measures. The measures of size of the input polyne mial are its degree d, coefficient length c, number of variables ra, and for sparse polynomials, the number of non-zero coefficients s. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. , zn) =įactoring Chanderjit Bajaj Rational * John Polynomials Canny t over the Complexes Garrity $ Joe Warren § Thomas Abstract We give NC algorithms for determining the number and degrees of the absolute factors (factors irreducible over the complex numbers C) of a multivariate polynomial with rational coefficients. These methods rely on the fact that the connected components of a complex hypersurface P(zl. Finally, we discuss a method for obtaining an approximation to the coefficients of each factor whose running time is polynomial in the size of the original (dense) polynomial. The algorithm also works in random NC for polynomials represented by straight-line programs, provided the polynomial can be evaluated at integer points in NC. If n is fixed, or if the polynomial is dense, we give a deterministic NC algorithm. Warren, J.įactoring Chanderjit Bajaj Rational * John Polynomials Canny t over the Complexes Garrity $ Joe Warren § Thomas Abstract We give NC algorithms for determining the number and degrees of the absolute factors (factors irreducible over the complex numbers C) of a multivariate polynomial with rational coefficients. Factoring rational polynomials over the complexes Factoring rational polynomials over the complexesīajaj, C.