The Gasca-Maeztu Conjecture for n = 4

We consider planar GCn node sets, i.e., n-poised sets whose all n-fundamental polynomials are products of n linear factors. Gasca and Maeztu conjectured in 1982 that every such set possesses a maximal line, i.e., a line passing through n + 1 nodes of the set. Till now the conjecture is confirmed to be true for n ≤ 5. The case n = 5 was proved recently by H. Hakopian, K. Jetter, and G. Zimmermann (Numer. Math. 127 (2014) 685{713). In this paper we bring a short and simple proof of the conjecture for n = 4.