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 number 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 or ±Tn (with Tn the Chebychev polynomial), and if the composition a ◦ b of polynomials a, b is the k 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 varieties invariant 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).