Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,k)+3= (n+1)+n+\cdots+(n-k+5)+3$ and $4 \le k\le n-1.$ In this paper we prove that there are at most seven linearly independent curves of degree less than or equal to $k$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly seven such curves. Namely, we prove that then the set $\mathcal X$ has a very special construction: all its nodes but three belong to a (maximal) curve of degree $k-3.$ Let us mention that in a series of such results this is the third one. In the end, an important application to the bivariate polynomial interpolation is provided, which is essential also for the study of the Gasca-Maeztu conjecture.

翻译：让一组节点 $\ mathccdots+(n- k+5)+3美元 和 4\ le k\le n-1美元 。 在本文件中,我们证明,最多有7个线性独立曲线,其程度小于或等于美元,通过所有节点的美元X。当完全有7个这样的曲线时,我们提供了对案例的定性。也就是说,我们证明,当时设定的 $\ mathcdots+(n+1)+nçcdots+(n-k+5)+3美元和4\le k\le n-1美元非常特殊的构造:所有节点但有3个都属于一个(maximal) 水平 $K-3 的曲线。让我们在本文中提及,在一系列结果中,这是第三个。在最后,提供了对双变量多数值间断层的重要应用,这也对气体研究至关重要。