  We obtain the $L^p$ decay of oscillatory integral operators $T_\lambda$ with
certain homogeneous polynomial phase of degree $d$ in $(n+n)$-dimensions. In
this paper we require that $d>2n$. If $d/(d-n)<p<d/n$, the decay is sharp and
the decay rate is related to the Newton distance. In the case of $p=d/n$ or
$d/(d-n)$, we also obtain the almost sharp decay, here "almost" means the decay
contains a $\log(\lambda)$ term. For otherwise, the $L^p$ decay of $T_\lambda$
is also obtained but not sharp. A counterexample also arises in this paper to
show that $d/(d-n)\leq p\leq d/n$ is not necessary to guarantee the sharp
decay.
