Polynomial Interpolation in Higher Dimension: From Simplicial Complexes to GC Sets

Geometrically characterized (GC) sets were introduced by Chung and Yao in their work on polynomial interpolation in $\mathbb R^d$. Conjectures on the structure of GC sets have been proposed by Gasca and Maeztu for the planar case, and in higher dimension by de Boor and by Apozyan and Hakopian. We investigate GC sets in dimension three or higher, and show that one way to obtain such sets is from the combinatorics of simplicial complexes.