Simultaneously preperiodic points for families of polynomials in normal form

Abstract:

Let $d>m>1$ be integers, let $c_1,\dots , c_{m+1}$ be distinct complex numbers, and let $\mathbf {f}(z):=z^d+t_1z^{m-1}+t_2z^{m-2}+\cdots + t_{m-1}z+t_m$ be an $m$-parameter family of polynomials. We prove that the set of $m$-tuples of parameters $(t_1,\dots , t_m)\in \mathbb {C}^m$ with the property that each $c_i$ (for $i=1,\dots , m+1$) is preperiodic under the action of the corresponding polynomial $\mathbf {f}(z)$ is contained in finitely many hypersurfaces of the parameter space $\mathbb {A}^m$.