Dynamics of split polynomial maps: uniform bounds for periods and applications

Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the Chebyshev polynomial of degree d. Let m and n be integers greater than 1, we prove that there exists an effectively computable constant c(m,n) depending only on m and n such that the following holds. Let f_1,…,f_n be polynomials with coefficients in K, which are disintegrated polynomials of degree at most m and let F be the induced coordinate-wise self-map of the n-th dimensional affine space, i.e. F(x_1,…,x_n)=(f_1(x_1),…,f_n(x_n)). Then the period of every irreducible F-periodic subvariety of the n-th dimensional affine space with non-constant projection to each coordinate axis is at most c(m,n). As an immediate application, we prove an instance of the dynamical Mordell-Lang problem following recent work of Xie. The main technical ingredients are Medvedev-Scanlon classification of invariant subvarieties together with classical and more recent results in Ritt’s theory of polynomial decomposition.