2018/07/21 by Hakopian, Hakop, Vardanyan, Vahagn
#41A05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.08182
A planar node set \mathcal X, with |\mathcal X|=\binomn+22 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 Ax+By+C=0 if Ax+By+C divides the fundamental polynomial of the 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 GCn sets and an (n+1)-node line is called 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 adjust and prove a conjecture proposed in the paper - V. Bayramyan, H. H., Adv Comput Math, 43: 607-626, 2017. 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 \binoms2 nodes of \mathcal X, where s satisfies the condition σ:=2k-n-1≤ s≤ k. If in addition σ≥ 3 and μ(\mathcal X)>3 then the first case here is excluded, i.e., the line ℓ is necessarily a used line. Here μ(\mathcal X) denotes the number of maximal lines of \mathcal X. At the end, we bring a characterization for the usage of k-node lines in GCn sets when σ=2 and μ(\mathcal X)>3.