On maximal curves of $n$-correct sets

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 $d(n,k)=\left(\genfrac{}{}{0ex}{}{n+2}{2}\right)-\left(\genfrac{}{}{0ex}{}{n+2-k}{2}\right).$ 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.

20 pages, 2 figures