On maximal curves of $n$-correct sets

TL;DR

This paper investigates the properties of maximal algebraic curves passing through the maximum number of nodes in n-correct sets, providing new insights and extensions to existing knowledge in polynomial interpolation.

Contribution

The paper introduces new properties and extends known results regarding maximal curves in n-correct node sets, enhancing understanding of their structure and significance.

Findings

Maximal curves pass through the maximum number of nodes as defined by d(n,k).

New properties of maximal curves are established, extending previous results.

Maximal lines pass through n+1 nodes, serving as a fundamental case.

Abstract

Suppose $\mathcal{X}$ is an $n$-correct set of nodes in the plane, that is, it admits a unisolvent interpolation with bivariate polynomials of total degree less than or equal to $n.$ Then an algebraic curve $q$ of degree $k\le n$ can pass through at most $d(n,k)$ nodes of $\Xset,$ where $d(n,k)={{n+2}\choose {2}}-{{n+2-k}\choose {2}}.$ A curve $q$ of degree $k\le n$ is called maximal if it passes through exactly $d(n,k)$ nodes of $\mathcal{X}.$ In particular, a maximal line is a line passing through $d(n,1)=n+1$ nodes of $\mathcal{X}.$ Maximal curves are an important tool for the study of $n$-correct sets. We present new properties of maximal curves, as well as extensions of known properties.