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

Abstract

An n-correct node set X is called GCn set if the fundamental polynomial of each node is a product of n linear factors. In 1982 Gasca and Maeztu conjectured that for every GCn set there is a line passing through n + 1 of its nodes. So far, this conjecture has been confirmed only for n ≤ 5. The case n = 4, was first proved by J. R. Busch [3]. 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 give a second proof, which largely follows the first one but is much shorter and simpler.