  To the best of our knowledge this paper is the first attempt to introduce and
study polynomial interpolation of the polynomial data given on arbitrary
varieties. In the first part of the paper we present results on the solvability
of such problems. In the second part of the paper we relate the interpolation
problem to polynomial solution of some boundary values problems. In particular,
we extend a result of W. K. Hayman and Z. G. Shanidze.
