On the usage of lines in GC_n sets

Abstract

A planar node set $\mathcal X,$ with $|\mathcal X|=\binom {n+2}{2},$ is called GC_n set if each node possesses fundamental polynomial in form of a product of n linear factors. We say that a node uses a line if the line is a component of the fundamental polynomial of this node. A line is called k-node line if it passes through exactly k nodes of $\mathcal X$. At most, n + 1 nodes can be collinear in any GC_n set and an (n + 1)-node line is called a maximal line. The Gasca-Maeztu conjecture (1982) states that every GC_n set has a maximal line. Until now, the conjecture has been proved only for the cases n ≤ 5. Here, we provide a correct statement and prove a conjecture proposed by V. Bayramyan and H. H. in a recent paper. Namely, by assuming that the Gasca-Maeztu conjecture is true, we prove that for any GC_n set $\mathcal X$ and any k-node line ℓ the following statement holds: Either the line ℓ is not used at all, or it is used by exactly $\binom {s}{2}$ nodes of $\mathcal X,$ where k − δ ≤ s ≤ k, δ = n + 1 − k. If in addition k − δ ≥ 3 and the number of maximal lines of the set $\mathcal X$ is greater than 3, then the first case here is excluded, i.e., ℓ necessarily is a used line. Finally, we provide a characterization for the usage of a k-node line in a GC_n set $\mathcal {X}$ concerning the case k − δ = 2.