  We consider a class of $0$-$1$ polynomial programming termed multiple choice
polynomial programming (MCPP) where the constraint requires exact one component
per subset of the partition to be $1$ after all the entries are partitioned.
Compared to the unconstrained counterpart, there are few polyhedral studies of
MCPP in general form. This paper serves as the first attempt to propose a
polytope associated with a hypergraph to study MCPP, which is the convex hull
of $0$-$1$ vectors satisfying multiple choice constraints and production
constraints. With the help of the decomposability property, we obtain an
explicit half-space representation of the MCPP polytope when the underlying
hypergraph is $\alpha$-acyclic by induction on the number of hyperedges, which
is an analogy of the acyclicity results on the multilinear polytope by Del Pia
and Khajavirad (SIAM J Optim 28 (2018) 1049) when the hypergraph is
$\gamma$-acyclic. We also present a necessary and sufficient condition for the
inequalities lifted from the facet-inducing ones for the multilinear polytope
to be still facet-inducing for the MCPP polytope. This result covers the
particular cases by B\"armann, Martin and Schneider (SIAM J Optim 33 (2023)
2909).
