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arXiv · 2507.11207

On maximal curves of $n$-correct sets

H. Hakopian·G. Vardanyan·N. Vardanyan
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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.

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H. Hakopian, G. Vardanyan, N. Vardanyan. 2025-07-15. On maximal curves of $n$-correct sets. https://arxiv.org/abs/2507.11207

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