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Abstract <p>It has been shown that for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5143_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(k=6l\pm 1\)</EquationSource> <!--CMatCMGU2470029Voronenko-m1--> </InlineEquation>, product <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5143_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(xy\)</EquationSource> <!--CMatCMGU2470029Voronenko-m2--> </InlineEquation> is a universal function for the class of linear functions of two variables. This work proves there is no universal polynomial for the class of linear functions of two variables for a <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5143_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\)</EquationSource> <!--CMatCMGU2470029Voronenko-m3--> </InlineEquation> multiple of three, or for the class of linear functions of three variables for even <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5143_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\)</EquationSource> <!--CMatCMGU2470029Voronenko-m4--> </InlineEquation>. This establishes a necessary and sufficient condition for the existence of a universal polynomial for the class of linear functions.</p>

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Criterion for the Existence of Universal Polynomials for the Class of Linear Functions

  • 期刊论文

  • 发表日期：

    2025年3月29日

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  • A. A. Voronenko,
  • A. S. Sedova

摘要

Abstract

It has been shown that for \(k=6l\pm 1\) , product \(xy\) is a universal function for the class of linear functions of two variables. This work proves there is no universal polynomial for the class of linear functions of two variables for a \(k\) multiple of three, or for the class of linear functions of three variables for even \(k\) . This establishes a necessary and sufficient condition for the existence of a universal polynomial for the class of linear functions.