  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.
