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