On the dimension of spaces of algebraic curves passing through $n$-independent nodes

Abstract

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 $d(n,k-2) = (n+1)+n+\cdots+(n-k+4)$ and $\ k\le n-1.$ 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.