ON $n$-NODE LINES IN $GC_n$ SETS

Abstract

An $n$-poised node set $\mathcal X$ in the plane is called $GC_n$ set, if the fundamental polynomial of each node is a product of linear factors. 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 $\Xset$ and an $(n+1)$-node line is called maximal line. The well-known conjecture of M. Gasca and J.I. Maeztu states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5.$ In this paper we prove some results concerning $n$-node lines, assuming that the Gasca--Maeztu conjecture is true.