Decomposition of polynomials and approximate roots

Abstract

We state a kind of Euclidian division theorem: given a polynomial P (x) and a divisor d of the degree of P , there exist polynomials h(x), Q(x), R(x) such that P (x) = h ◦ Q(x) + R(x), with deg h = d. Under some conditions h, Q, R are unique, and Q is the approximate d-root of P . Moreover we give an algorithm to compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not.