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Abstract <p>Suppose <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathcal{X}}\)</EquationSource> <!--ContMath2570021Hakopian-m3--> </InlineEquation> is an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--ContMath2570021Hakopian-m4--> </InlineEquation>-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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(n.\)</EquationSource> <!--ContMath2570021Hakopian-m5--> </InlineEquation> Then an algebraic curve <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(q\)</EquationSource> <!--ContMath2570021Hakopian-m6--> </InlineEquation> of degree <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(k\leq n\)</EquationSource> <!--ContMath2570021Hakopian-m7--> </InlineEquation> can pass through at most <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(d(n,k):={{n+2}\choose{2}}-{{n+2-k}\choose{2}}\)</EquationSource> <!--ContMath2570021Hakopian-m8--> </InlineEquation> nodes of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\({\mathcal{X}}.\)</EquationSource> <!--ContMath2570021Hakopian-m9--> </InlineEquation> A curve <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(q\)</EquationSource> <!--ContMath2570021Hakopian-m10--> </InlineEquation> of degree <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(k\leq n\)</EquationSource> <!--ContMath2570021Hakopian-m11--> </InlineEquation> is called maximal if it passes through exactly <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(d(n,k)\)</EquationSource> <!--ContMath2570021Hakopian-m12--> </InlineEquation> nodes of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\({\mathcal{X}}.\)</EquationSource> <!--ContMath2570021Hakopian-m13--> </InlineEquation> In particular, a maximal line is a line passing through <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(d(n,1)=n+1\)</EquationSource> <!--ContMath2570021Hakopian-m14--> </InlineEquation> nodes of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\({\mathcal{X}}.\)</EquationSource> <!--ContMath2570021Hakopian-m15--> </InlineEquation> Maximal curves are an important tool for the study of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--ContMath2570021Hakopian-m16--> </InlineEquation>-correct sets. We present new properties of maximal curves, as well as extensions of known properties.</p>

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On Maximal Curves of \(\boldsymbol{n}\)-Correct Sets

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • H. A. Hakopian,
  • G. K. Vardanyan,
  • N. K. Vardanyan

摘要

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 \(k\leq n\) can pass through at most \(d(n,k):={{n+2}\choose{2}}-{{n+2-k}\choose{2}}\) nodes of \({\mathcal{X}}.\) A curve \(q\) of degree \(k\leq n\) 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.