On maximal curves of $n$-correct sets
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908450456338432 |
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| author | Hakopian, H. Vardanyan, G. Vardanyan, N. |
| author_facet | Hakopian, H. Vardanyan, G. Vardanyan, N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On maximal curves of $n$-correct sets Hakopian, H. Vardanyan, G. Vardanyan, N. Numerical Analysis Algebraic Geometry 41A05, 41A63, 14H50 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. |
| title | On maximal curves of $n$-correct sets |
| topic | Numerical Analysis Algebraic Geometry 41A05, 41A63, 14H50 |
| url | https://arxiv.org/abs/2507.11207 |