On the Gasca-Maeztu conjecture for $n=6$

TL;DR

This paper investigates the Gasca-Maeztu conjecture for the case n=6, focusing on the structure of correct node sets with linear fundamental polynomials in bivariate interpolation.

Contribution

It advances the proof of the Gasca-Maeztu conjecture specifically for n=6, extending previous results confirmed for n≤5.

Findings

Confirmed the conjecture for n=6 under certain conditions

Identified structural properties of correct node sets for n=6

Extended the understanding of collinearity in bivariate interpolation sets

Abstract

A two-dimensional $n$-correct set is a set of nodes admitting unique bivariate interpolation with polynomials of total degree at most ~$n$. We are interested in correct sets with the property that all fundamental polynomials are products of linear factors. In 1982, M.~Gasca and J.~I.~Maeztu conjectured that any such set necessarily contains $n+1$ collinear nodes. So far, this had only been confirmed for $n\leq 5.$ In this paper, we make a step for proving the case $n=6.$