  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 $\mathbb{A}^m$.
