The Gasca–Maeztu conjecture for $$n=5$$ n = 5

Abstract

An $$n$$ n -poised set in two dimensions is a set of nodes admitting unique bivariate interpolation with polynomials of total degree at most $$n$$ n . We are interested in poised sets with the property that all fundamental polynomials are products of linear factors. Gasca and Maeztu (Numer Math 39:1---14, 1982) conjectured that every such set necessarily contains $$n+1$$ n + 1 collinear nodes. Up to now, this had been confirmed only for $$n\le 4$$ n ≤ 4 , the case $$n=4$$ n = 4 having been proved for the first time by Busch (Rev Un Mat Argent 36:33---38, 1990). In the present paper, we prove the case $$n=5$$ n = 5 with new methods that might also be useful in deciding the still open cases for $$n\ge 6$$ n ? 6 .