Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textit{k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="italic">k</mi> </math></EquationSource> </InlineEquation> be an algebraically field of characteristic <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$a\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and X a smooth projective superelliptic curve given by affine equation of the form <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$y^a =f(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mi>a</mi> </msup> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Let denoted by b the degree of f and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\textit{K}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">K</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the function field of X. For some smooth projective superelliptic curves X of genus <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$g_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation>, we give an explicit <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textit{ k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="4pt" /> <mi mathvariant="italic">k</mi> </mrow> </math></EquationSource> </InlineEquation>-basis of the space of global holomorphic polydifferentials <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2515_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$H^0 ( X, \Omega ^{\otimes m}_{X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mi>X</mi> <mrow> <mo>⊗</mo> <mi>m</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of order m.

On the Space of Polydifferentials Forms on Some Superelliptic Curves

期刊论文
发表日期：

2025年9月12日

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Hilaire George Mbiakop,
Celestin Jugnia Nkuimi,
Alex Stephane Pokam

摘要

Let $\textit{k}$ $k$ be an algebraically field of characteristic $p\ne 2$ $p \neq 2$ . Let $a\ge 2$ $a \geq 2$ and X a smooth projective superelliptic curve given by affine equation of the form $y^a =f(x)$ $y^{a} = f (x)$ . Let denoted by b the degree of f and $\textit{K}(X)$ $K (X)$ the function field of X. For some smooth projective superelliptic curves X of genus $g_X$ $g_{X}$ , we give an explicit $\textit{ k}$ $k$ -basis of the space of global holomorphic polydifferentials $H^0 ( X, \Omega ^{\otimes m}_{X})$ $H^{0} (X, Ω_{X}^{\otimes m})$ of order m.