Polynomial interpolation in higher dimension: from simplicial complexes to GC sets
Provides a theoretical advance for mathematicians working on multivariate polynomial interpolation by linking GC sets to simplicial complexes.
The paper investigates geometrically characterized (GC) sets for polynomial interpolation in higher dimensions, showing that such sets can be obtained from simplicial complexes, addressing conjectures by Gasca-Maeztu, de Boor, and Apozyan-Hakopian.
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.