On the basic properties of $GC_n$ sets (2001.05306v3)

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 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 $GC_n$ 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 $n \le 5.$ Here, for a line $\ell$ we introduce and study the concept of $\ell$-lowering of the set $\mathcal X$ 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.