2020/01/13 by Hakopian, Hakop, Vardanyan, Navasard · 2 citations
#41A05 #41A63 #Combinatorics (math.CO) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2001.05306
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 if the line is a factor of 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 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, for a line ℓ we introduce and study the concept of ℓ-lowering of the set \mathcal X and define so called proper lines. We also provide refinements of several basic properties of GCn sets regarding the maximal lines, n-node lines, the used lines, as well as the subset of nodes that use a given line.