ON RITT'S POLYNOMIAL DECOMPOSITION THEOREMS

Ritt studied the functional decomposition of a univariate complex polynomial f into prime (indecomposable) polynomials, f = u1 ◦ u2 ◦ � � � ◦ ur. His main achievement was a procedure for obtaining any decomposition of f from any other by repeatedly applying certain transformations. However, Ritt's results provide no control on the num- ber of times one must apply the basic transformations, which makes his procedure unsuitable for many theoretical and algorithmic applications. We solve this problem by giving a new description of the collection of all decompositions of a polynomial. One consequence is as follows: if f has degree n > 1 but f is not conjugate by a linear polynomial to either X n or ±Tn (with Tn the Chebychev polynomial), and if the composition a ◦ b of polynomials a, b is the k th iterate of f for some k > log2(n + 2), then either a = f ◦ c or b = c ◦ f for some polynomial c. This result has been used by Ghioca, Tucker and Zieve to describe the polynomials f, g having orbits with infinite intersection; our results have also been used by Medevedev and Scanlon to describe the affine curves invari- ant under a coordinatewise polynomial action. Ritt also proved that the sequence (deg(u1), . . . ,deg(ur)) is uniquely determined by f, up to permutation. We show that in fact, up to permutation, the sequence of permutation groups (G(u1), . . . , G(ur)) is uniquely determined by f, where G(u) = Gal(u(X)−t, C(t)). This generalizes both Ritt's invariant and an invariant discovered by Beardon and Ng, which turns out to be equivalent to the subsequence of cyclic groups among the G(ui).

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