On the usage of $2$-node lines in $n$-correct and $GC_n$ sets

Abstract

An $n$-correct set $\mathcal{X}$ in the plane is a set of nodes admitting unique interpolation with bivariate polynomials of total degree at most $n$. A $k$-node line is a line passing through exactly $k$ nodes of $\mathcal{X}.$ A line can pass through at most $n+1$ nodes of an $n$-correct set. An $(n+1)$-node line is called maximal line (C. de Boor, 2007). We say that a node $A\in\mathcal{X}$ uses a line $\ell,$ if $\ell$ is a factor of the fundamental polynomial of the node $A.$ Let $\mathcal{X}$ be an $n$-correct set. One of the main problems we study in this paper is to determine the maximum possible number of used $2$-node lines that share a common node $B \in\mathcal{X}.$ We show that this number equals $n$. Moreover, if there are $n$ such $2$-node lines, then $\mathcal{X}$ contains exactly $n$ maximal lines not passing through the common node $B$. Furthermore, if $\mathcal{X}$ is $GC_n$ set, there exists an additional maximal line passing through $B$. Hence, in this case, $\mathcal{X}$ has $n+1$ maximal lines and is Carnicer~Gasca set of degree $n$. Note that Carnicer~Gasca sets of degree $n$ with a prescribed set of $n$ used $2$-node lines can be readily constructed.