On plane algebraic curves passing through $n$-independent nodes

TL;DR

This paper investigates the maximum number of low-degree algebraic curves passing through a specially constructed set of plane nodes, providing a characterization of cases with the maximum number and applying results to polynomial interpolation and the Gasca-Maeztu conjecture.

Contribution

It establishes an upper bound of seven on the number of such curves and characterizes the configuration when this maximum is achieved, advancing understanding in algebraic geometry and polynomial interpolation.

Findings

Maximum of seven linearly independent curves of degree ≤ k passing through the nodes.

Characterization of node sets when exactly seven such curves exist.

Application to bivariate polynomial interpolation and implications for the Gasca-Maeztu conjecture.

Abstract

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 $\#\mathcal X=d(n,k-3)+3= (n+1)+n+\cdots+(n-k+5)+3$ 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.