On the usage of lines in $G{C}_n$ sets

Abstract

A planar node set $\mathcal{X},$ with $|\mathcal{X}|=\left(\genfrac{}{}{0ex}{}{n+2}{2}\right)$ is called $G{C}_n$ set if each node possesses fundamental polynomial in form of a product of $n$ linear factors. We say that a node uses a line $Ax+By+C=0$ 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 $G{C}_n$ sets and an $(n+1)$-node line is called maximal line. The Gasca - Maeztu conjecture (1982) states that every $G{C}_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n\le 5.$ 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 $G{C}_n$ set $\mathcal{X}$ and any $k$-node line $\ell$ the following statement holds: Either the line $\ell$ is not used at all, or it is used by exactly $\left(\genfrac{}{}{0ex}{}{s}{2}\right)$ nodes of $\mathcal{X},$ where $s$ satisfies the condition $\sigma :=2k-n-1\le s\le k.$ If in addition $\sigma \ge 3$ and $\mu (\mathcal{X})>3$ then the first case here is excluded, i.e., the line $\ell$ is necessarily a used line. Here $\mu (\mathcal{X})$ denotes the number of maximal lines of $\mathcal{X}.$ At the end, we bring a characterization for the usage of $k$-node lines in $G{C}_n$ sets when $\sigma =2$ and $\mu (\mathcal{X})>3.$

24 pages, 6 figures. arXiv admin note: text overlap with arXiv:1712.06155