ON THE DIMENSION OF SPACES OF ALGEBRAIC CURVESPASSING THROUGH n-INDEPENDENT NODES

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Submitted: 2025-01-30; Published: 2025-01-30

This work is licensed under Creative Commons Attribution–NonCommercial International License (CC BY-NC 4.0).

Abstract

Let the set of nodesXin the plain ben-independent, i.e., each node has afundamental polynomial of degreen.Suppose also that|X|= (n+1)+n+···++(n−k+4)+2 and 3≤k≤n−1.We prove that there can be at most 4 line-arly independent curves of degree less than or equal tokpassing through all thenodes ofX.We provide a characterization of the case when there are exactly 4such curves. Namely, we prove that then the setXhas a very special construc-tion: all its nodes but two belong to a (maximal) curve of degreek−2.At theend, an important application to the Gasca-Maeztu conjecture is provided.