Functional decomposition of polynomials

Fast DecoIIlposition in the tame case 2 polynomials over F. We obtain a range of results, trom Ulldecidability over sufficiently general fields to fast sequential and parallel algorithms over finite fields. A version of the algorithm of Theorem 1 below has beel implemented [2,6J and compares favorably with [3J. Dick erson [9J has extended some of these results to multivariate polynomials. We should give a brief history of the research behind this joint paper. Kozen and Landau [18] gave the first polynomial-time sequential and NCalgorithms for this problem in the tame case. The time hounds were O(n3 ) sequential, O(n ) if F supports an FFT, and 0(1og2 n) parallel. They also presented the structure theorem (Theorem 9), reducing the problem in the wild case to factorization, and gave an O(n ) algorithm for the decomposition of irreducible polynomials over general fields admitting a polynomial-time factorization algorithm, and an NC algorithm for irreducible polynomials over finite fields. Based on the algorithm of [18], von zur Gathen [17] improved the bounds in the tame case to those stated above. These results are presented in §2. He also gave an improved algorithm for the wild case, yielding a polynomial-time reduction to factorization of polynomials, and observed undecidability over sufficiently general fields. These results are presented in §3. Introduction 1