Simultaneously preperiodic points for families of polynomials in normal form

Let d > m > 1 be integers, let c₁, …, c_m+1 be distinct complex numbers, and let f(z):= z^d + t₁ z^m−1 + t₂ z^m−2 + · · · + t_m−1 z + t_m be an m-parameter family of polynomials. We prove that the set of m-tuples of parameters (t₁, …, t_m) ∈ C^m with the property that each c_i (for i = 1, …, m+1) is preperiodic under the action of the corresponding polynomial f(z) is contained in finitely many hypersurfaces of the parameter space A^m.