Polynomial interpolation in higher dimension: from simplicial complexes to GC sets

TL;DR

This paper explores the structure of GC sets in higher dimensions, linking their formation to the combinatorics of simplicial complexes, and advances understanding of polynomial interpolation in multi-dimensional spaces.

Contribution

It introduces a new method to construct GC sets in higher dimensions using simplicial complexes, extending prior work focused mainly on the planar case.

Findings

GC sets in higher dimensions can be derived from simplicial complex combinatorics

The work extends the understanding of polynomial interpolation in R^d

Provides a framework for constructing GC sets beyond the plane

Abstract

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