On a correction of a property of GCn sets

Abstract

An n-poised node set $\mathcal {X}$ in the plane is called GCn 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 a maximal line. In 1982, Gasca and Maeztu conjectured that every GCn set has a maximal line. Until now the conjecture has been proved only for n ≤ 5. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. It is a simple fact that any maximal line λ is used by all $\binom {n + 1}{2}$ nodes in $\mathcal {X} \setminus \lambda $ . We consider the main result of the paper—Bayramyan and Hakopian (Adv. Comput. Math. 43, 607–626, 2017) stating that any n-node line of GCn 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. Here, we show that this result is not correct in the case n = 3. Namely, we bring an example of a GC3 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 abovementioned result is true for all n ≥ 4. 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 GCn sets.