Factoring Rational Polynomials Over the Complex Numbers

NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers C, of a multivariate polynomial with rational coeÆcients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree d, coeÆcient length c, number of variables n. If n is xed, we give a deterministic NC algorithm. If the number of variables is not xed, we give a random (Monte-Carlo) NC algorithm in these input measures to nd the number and degree of each irreducible factor. After reducing to the two variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of (P (z1; z2) = 0) correspond to the connected components of the real surface (or complex curve) P (z1; z2) = 0 minus its singular points. In nding the number of connected components of the surface P = 0, Department of Computer Science, Purdue University. Supported in part by ARO Contract DAAG29-85-C0018 and ONR contract N00014-88-K-0402 Computer Science Division, Berkeley. Supported in part by a David and Lucille Packard Fellowship Department of Mathematics, Rice University Department of Computer Science, Rice University. Supported in part by NSF grant IRI 88-10747