Simultaneously preperiodic points for families of polynomials in normal form

Abstract

Let d > m > 1 be integers, let c 1 , . . . , c m+1 be distinct complex numbers, and let f (z) := z d + t 1 z m−1 + t 2 z m−2 + · · · + t m−1 z + tm be an mparameter family of polynomials. We prove that the set of m-tuples of parameters (t 1 , . . . , tm) ∈ 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 .