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<p>We give an overview on the landscape of polynomial interpolation theory. We will describe first the geometric approach, based on the base locus analysis of linear systems of hypersurfaces of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_468_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {P}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with given degree and assigned multiplicity at a set of points. Secondly, we will consider the algebraic counterpart, with a discussion on the good postulation of fat point schemes of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_468_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {P}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and their regularity index. In both cases we report on some complete, or partial, results and conjectures.</p>

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Towards Good Postulation of Fat Points, One Step at a Time

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月2日

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  • Maria Chiara Brambilla,
  • Elisa Postinghel

摘要

We give an overview on the landscape of polynomial interpolation theory. We will describe first the geometric approach, based on the base locus analysis of linear systems of hypersurfaces of \({\mathbb {P}}^n\) P n with given degree and assigned multiplicity at a set of points. Secondly, we will consider the algebraic counterpart, with a discussion on the good postulation of fat point schemes of \({\mathbb {P}}^n\) P n and their regularity index. In both cases we report on some complete, or partial, results and conjectures.