A new proof of the Gasca-Maeztu conjecture for n = 5

TL;DR

This paper presents a shorter, more accessible proof of the Gasca-Maeztu conjecture for the case n=5, confirming the existence of a line passing through six nodes in any GC_5 set.

Contribution

The paper offers a new, simplified proof of the Gasca-Maeztu conjecture for n=5, improving upon the previous complex proof by Hakopian, Jetter, and Zimmermann.

Findings

Confirmed the Gasca-Maeztu conjecture for n=5

Provided a shorter proof compared to previous work

Strengthened understanding of GC_n sets for n=5

Abstract

An $n$-correct node set $\mathcal{X}$ is called $GC_n$ set if the fundamental polynomial of each node is a product of $n$ linear factors. In 1982 Gasca and Maeztu conjectured that for every $GC_n$ set there is a line passing through $n+1$ of its nodes.So far, this conjecture has been confirmed only for $n\le 5.$ The case $n = 4,$ was first proved by J. R. Bush in 1990. Several other proofs have been published since then. For the case $n=5$ there is only one proof: by H. Hakopian, K. Jetter and G. Zimmermann (Numer Math $127,685-713, 2014$). Here we present a second, much shorter and easier proof.