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Composition factors of polynomials

A complex polynomial is composite if it can be expressed as the composition of two nonlinear polynomials; otherwise it is prime. It is easy to see that any polynomial p can be expressed as a composition, say , where the p j are prime. This decomposition into prime polynomials is not unique; however, in 1926 Ritt showed that the set of degrees of the p j is uniquely determined by p. Here we show that the vector of degrees determines the p j uniquely. The critical points of p are the solutions of P′(z)=0, and the critical values of p are the images, under p, of its critical points. We give a preliminary investigation into the role of critical points and critical values in the possible factorizations of p, and we apply our results to polynomials with large gaps in their coefficients, and to the perturbations of a polynomial.