Multivariate polynomial interpolation: conjectures concerning GC-sets

Abstract

GC-sets are subsets T of \(\mathbb{R}^d\) of the appropriate cardinality \(\dim\Pi_n\) for which, for each τ ∈ T, there are n hyperplanes whose union contains all of T except for τ, thus making interpolation to arbitrary data on T by polynomials of degree ≤ n uniquely possible. The existing bivariate theory of such sets is extended to the general multivariate case and the concept of a maximal hyperplane for T is highlighted, in hopes of getting more insight into existing conjectures for the bivariate case.