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<p>We consider a&#xa0;general system of transcendental equations inthe <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1518_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$ n $</EquationSource> </InlineEquation>-dimensional complex space<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1518_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ 𝔺^{n} $</EquationSource> </InlineEquation>and introduce the concept of&#xa0;<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1518_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \sigma $</EquationSource> </InlineEquation>-power sum of the rootsof a&#xa0;system&#xa0;<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1518_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$ \sigma_{\alpha} $</EquationSource> </InlineEquation>.Multidimensional residue theoryyields the formulas that relatethe <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1518_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \sigma $</EquationSource> </InlineEquation>-power sums of various ordersto each other.We establish a&#xa0;connection between the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1518_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \sigma $</EquationSource> </InlineEquation>-power sumsand the power sums of the roots of the system.</p>

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Analogs of Recurrent Newton Formulas for Systems of Transcendental Equations

  • 期刊论文

  • 发表日期：

    2025年1月29日

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  • A. M. Kytmanov

摘要

We consider a general system of transcendental equations inthe $ n $ -dimensional complex space $ 𝔺^{n} $ and introduce the concept of  $ \sigma $ -power sum of the rootsof a system  $ \sigma_{\alpha} $ .Multidimensional residue theoryyields the formulas that relatethe $ \sigma $ -power sums of various ordersto each other.We establish a connection between the $ \sigma $ -power sumsand the power sums of the roots of the system.