Abstract
A planar node set \(\mathcal X,\) with \(|\mathcal X|=\binom {n+2}{2},\) is called GCn 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 GCn set and an (n + 1)-node line is called a maximal line. The Gasca-Maeztu conjecture (1982) states that every GCn 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 GCn 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 GCn set \(\mathcal {X}\) concerning the case k − δ = 2.
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We thank reviewers for thorough reviews and appreciate comments, suggestions, and corrections that have made a significant contribution to improving the quality of the paper.
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Communicated by: Larry L. Schumaker
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Hakopian, H., Vardanyan, V. On the usage of lines in GCn sets. Adv Comput Math 45, 2721–2743 (2019). https://doi.org/10.1007/s10444-019-09705-w
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DOI: https://doi.org/10.1007/s10444-019-09705-w