Function theory on the Neil parabola

We give a formula for the Carath'eodory distance on the Neil parabola, the variety $z^2=w^3$ restricted to the bidisk; thus making it the first variety with a singularity to have its Carath'eodory distance explicitly computed. In addition, we relate this to a mixed Carath'eodory-Pick interpolation problem for which known interpolation theorems do not apply. Finally, we prove a bounded holomorphic function extension result from the Neil parabola to the bidisk.