  We study the space of orthogonally additive $n$-homogeneous polynomials on
$C(K)$. There are two natural norms on this space. First, there is the usual
supremum norm of uniform convergence on the closed unit ball. As every
orthogonally additive $n$-homogeneous polynomial is regular with respect to the
Banach lattice structure, there is also the regular norm. These norms are
equivalent, but have significantly different geometric properties. We
characterise the extreme points of the unit ball for both norms, with different
results for even and odd degrees. As an application, we prove a Banach-Stone
theorem. We conclude with a classification of the exposed points.
