2016/02/10 by Vahagn Bayramyan, Bayramyan, Vahagn, Hakop Hakopian +1
Computer Science · Mathematics · #14H50 #41A05 #41A63 #Advanced Differential Equations and Dynamical Systems #Computational Geometry and Mesh Generation #FOS: Mathematics #Numerical Analysis (math.NA) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1602.03338
openalex publication_date 2016/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper we consider n-poised planar node sets, as well as more special ones, called GCn-sets. For these sets all n-fundamental polynomials are products of n linear factors as it always takes place 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 \mathcal X∖ M, meaning that it is a factor of the fundamental polynomial of each node. 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 \binomn2 nodes or by exactly \binomn-12 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.