  We prove the $L^p$-boundedness, $1<p<\infty$, of the Polynomial Carleson
operator in general dimension. This follows the author's resolution of the one
dimensional case as well as the work of Zorin-Kranich on the higher dimensional
case in the setting $2\leq p<\infty$. The techniques used in this paper are
direct adaptations and natural extensions to the higher dimensional case of the
one-dimensional methods developed by the author.
