On a new property of n-poised and G C n sets

Abstract

In this paper we consider n-poised planar node sets, as well as more special ones, called GCn sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line l is called k-node line for a node set ??$\mathcal X$ if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every GCn set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of ??$\mathcal X$ is used by each node in ???M,$\mathcal X\setminus M, $meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of GCn set ??$\mathcal {X}$ is used either by exactly n2$\binom {n}{2}$ nodes or by exactly n?12$\binom {n-1}{2}$ nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and GCn sets. At the end we present a conjecture concerning any k-node line.