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Abstract <p> For each of the forty-eight exceptional algebraic solutions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2685_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(u(x)\)</EquationSource> </InlineEquation> of the sixth Painlevé equation, we build the algebraic curve <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2685_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">\(P(u,x)=0\)</EquationSource> </InlineEquation> of a degree conjectured to be minimal, and then we give an optimal parametric representation of it. This degree is equal to the number of branches, except for fifteen solutions. </p>

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Minimal algebraic solutions of the sixth Painlevé equation

  • 期刊论文

  • 发表日期：

    2025年9月26日

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  • R. Conte

摘要

Abstract

For each of the forty-eight exceptional algebraic solutions \(u(x)\) of the sixth Painlevé equation, we build the algebraic curve \(P(u,x)=0\) of a degree conjectured to be minimal, and then we give an optimal parametric representation of it. This degree is equal to the number of branches, except for fifteen solutions.