  In this work we extend consideration of the polynomial Carleson operator to
the setting of a Radon transform acting along the paraboloid in
$\mathbb{R}^{n+1}$ for $n \geq 2$. Inspired by work of Stein and Wainger on the
original polynomial Carleson operator, we develop a method to treat polynomial
Carleson operators along the paraboloid via van der Corput estimates. A key new
step in the approach of this paper is to approximate a related maximal
oscillatory integral operator along the paraboloid by a smoother operator,
which we accomplish via a Littlewood-Paley decomposition and the use of a
square function. The most technical aspect then arises in the derivation of
bounds for oscillatory integrals involving integration over lower-dimensional
sets. The final theorem applies to polynomial Carleson operators with phase
belonging to a certain restricted class of polynomials with no linear terms and
whose homogeneous quadratic part is not a constant multiple of the defining
function $|y|^2$ of the paraboloid in $\mathbb{R}^{n+1}$.
