New algorithms for exact and approximate polynomial decomposition

Computing a decomposition of a polynomial f(x) as a functional composition g(h(x)) of polynomials g(x) and h(x), is an important and well-studied problem, both for exact and approximate inputs. In this paper, we re-examine the original (exponential-time) algorithm of Barton and Zippel for this task, which looks for special factors of an associated separated bivariate polynomial. We demonstrate algorithms using this approach which are reasonably fast (i.e., run in a polynomial number of operations) for exact computation, and provide an effective new approach for the decomposition of approximate polynomials. For approximate polynomials we exhibit rigorous lower bounds on the distance to the nearest decomposable polynomial, as well as robust numerical algorithms.