ON A CONJECTURE IN BIVARIATE INTERPOLATION

Abstract

Denote the space of all bivariate polynomials of total degree n$\leq n$ by $\Pi_n$. We are interested in n-poised sets of nodes with the property that the fundamental polynomial of each node is a product of linear factors. In 1981 M. Gasca and J. I.Maeztu conjectured that every such set contains necessarily $n+1$ collinear nodes. Up to now this had been confirmed for degrees $n \leq 5$. Here we bring a simple and short proof of the conjecture for $n = 4$.