2019/03/24 by Hakopian, Hakop, Kloyan, Harutyun
#14H50 #41A05 #41A63 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.10874
Let a set of nodes \mathcal X in plain be n-independent, i.e., each node has a fundamental polynomial of degree n. Suppose also that |\mathcal X|= d(n,k-2)+2, where d(n,k-2) = (n+1)+n+⋯+(n-k+4) and k≤ n-1. In this paper we prove that there can be at most 4 linearly independent curves of degree less than or equal to k passing through all the nodes of \mathcal X. We provide a characterization of the case when there are exactly four such curves. Namely, we prove that then the set \mathcal X has a very special construction: All its nodes but two belong to a (maximal) curve of degree k-2. At the end, an important application to the Gasca-Maeztu conjecture is provided.