A new proof of the Gasca–Maeztu conjecture for n=4n=4

In the Chung–Yao construction of poised nodes for bivariate polynomial interpolation [K.C. Chung, T.H. Yao, On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal. 14 (1977) 735–743], the interpolation nodes are intersection points of some lines. The Berzolari–Radon construction [L. Berzolari, Sulla determinazione di una curva o di una superficie algebrica e su alcune questioni di postulazione, Lomb. Ist. Rend. 47 (2) (1914) 556–564; J. Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948) 286–300] seems to be more general, since in this case the nodes of interpolation lie (almost) arbitrarily on some lines. In 1982 Gasca and Maeztu conjectured that every poised set allowing the Chung–Yao construction is of Berzolari–Radon type. So far, this conjecture has been confirmed only for polynomial spaces of small total degree n≤4n≤4, the result being evident for n≤2n≤2 and not hard to see for n=3n=3. For the case n=4n=4 two proofs are known: one of J.R. Busch [J.R. Busch, A note on Lagrange interpolation in R2R2, Rev. Un. Mat. Argentina 36 (1990) 33–38], and another of J.M. Carnicer and M. Gasca [J.M. Carnicer, M. Gasca, A conjecture on multivariate polynomial interpolation, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) Ser. A Mat. 95 (2001) 145–153]. Here we present a third proof which seems to be more geometric in nature and perhaps easier. We also present some results for the case of n=5n=5 and for general nn which might be useful for later consideration of the problem.