On the basic properties of GC_n sets

01/13/2020

∙

by   Hakop Hakopian, et al.

∙

0

∙

share

A planar node set X, with # X=n+22, 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 factor of the fundamental polynomial of the node. A line is called k-node line if it passes through exactly k-nodes of 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, for a line ℓ we introduce and study the concept of ℓ-lowering of the set X and define so called proper lines. We also provide refinements of several basic properties of GC_n sets regarding the maximal lines, n-node lines, the used lines, as well as the subset of nodes that use a given line.