On a correction of a property of $GC$ sets

TL;DR

This paper investigates properties of $GC_n$ sets, disproves a specific conjecture for $n=3$, and characterizes the unique counterexample, advancing understanding of node line usage in polynomial interpolation.

Contribution

It provides a counterexample for $n=3$ to a conjecture about line usage in $GC_n$ sets and establishes that the conjecture holds for all other $n$, along with new characterizations.

Findings

Counterexample for $n=3$ showing the conjecture does not hold

The conjecture is valid for all $n eq 3$

Characterization of the unique $n=3$ counterexample

Abstract

An $n$-poised node set $\mathcal X$ in the plane is called $GC_n$ set if the (bivariate) fundamental polynomial of each node is a product of n linear factors. A line is called $k$-node line if it passes through exactly $k$-nodes of $\mathcal X.$ An $(n+1)$-node line is called maximal line. The well-known conjecture of M. Gasca and J. I. Maeztu states that every $GC_n$ set has a maximal line. Untill now the conjecture has been proved only for the cases $n \le 5.$ We say that a node uses a line if the line is a factor in the node's fundamental polynomial. It is a simple and well-known fact that any maximal line $M$ is used by all $\binom{n+1}{2}$ nodes in $\mathcal X\setminus M.$ Here we consider the main result of the paper - V. Bayramyan, H. Hakopian, On a new property of n-poised and $GC_n$ sets, Adv Comput Math, 43, (2017) 607-626, stating that any $n$-node line of $GC_n$ set is used either by exactly $\binom{n}{2}$ nodes or by exactly $\binom{n-1}{2}$ nodes, provided that the Gasca-Maeztu conjecture is true. In this paper we show that this result is not correct in the case $n=3.$ Namely, we bring an example of a $GC_3$ set and a $3$-node line there which is not used at all. Fortunately, then we were able to establish that this is the only possible counterexample, i.e., the above mentioned result is true for all $n\ge 1, n\neq 3.$ We also characterize the exclusive case $n=3$ and present some new results on the maximal lines and the usage of $n$-node lines in $GC_n$ sets.