  We determine conditions on q for the nonexistence of deep holes of the
standard Reed-Solomon code of dimension k over F_q generated by polynomials of
degree k+d. Our conditions rely on the existence of q-rational points with
nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces
defined over F_q. We show that the hypersurfaces under consideration are
invariant under the action of the symmetric group of permutations of the
coordinates. This allows us to obtain critical information concerning the
singular locus of these hypersurfaces, from which the existence of q-rational
points is established.
