On the singularity of multivariate Hermite interpolation

In this paper, we study the singularity of multivariate Hermite interpolation of type total degree. We present two methods to judge the singularity of the interpolation schemes considered and by methods to be developed, we show that all Hermite interpolation of type total degree on m=d+km=d+k points in RdRd is singular if d≥2kd≥2k. And then we solve the Hermite interpolation problem on m≤d+3m≤d+3 nodes completely. Precisely, all Hermite interpolations of type total degree on m≤d+1m≤d+1 points with d≥2d≥2 are singular; only three cases for m=d+2m=d+2 and one case for m=d+3m=d+3 can produce regular Hermite interpolation schemes, respectively. Besides, we also present a method to compute the interpolation space for Hermite interpolation of type total degree.