Observations on interpolation by total degree polynomials in two variables

TL;DR

This paper investigates the algebraic properties of specific point configurations used in bivariate polynomial interpolation, focusing on the Radon-Berzolari and Chung-Yao methods, and explores their geometric implications.

Contribution

It provides a detailed algebraic analysis of interpolation configurations, using H-bases and syzygies to understand their geometric structure.

Findings

Properties of the matrix of first syzygies are derived.

Connections between algebraic properties and geometric configurations are established.

Insights into the limitations of total degree polynomial interpolation in two variables.

Abstract

In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon-Berzolari and Chung-Yao. By means of proper H-bases for the vanishing ideal of the configuration, we derive properties of the matrix of first syzygies of this ideal which allow us to draw conclusions on the geometry of the point configuration.