  In contrast to the univariate case, interpolation with polynomials of a given
maximal total degree is not always possible even if the number of interpolation
points and the space dimension coincide. Due to that, numerous constructions
for interpolation sets have been devised, the most popular ones being based on
intersections of lines. In this paper, we study algebraic properties of some
such interpolation configurations, namely the approaches by Radon-Berzolari and
Chung-Yao. By means of proper H-bases for the vanishing ideal of the
configuration, we derive properties of the matrix of first syzygies of this
ideal which allow us to draw conclusions on the geometry of the point
configuration.
