Singularity of bivariate interpolation (Q1191740)

The classical bivariate Hermite interpolation problem is considered. Its conditions can be described by the collection \(E\) of all corresponding incidence matrices, and G. and R. Lorentz posed the problem of complete characterization of singular matrices \(E\) in terms of the multiplicities of the Hermite conditions. In the present paper, infinitely many new singular matrices are constructed which correspond to the Hermite interpolation with equal multiplicities and a geometrical interpretation is shown. Furthermore the general case of arbitrary multiplicities is discussed and sufficient conditions for singularity are given.