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Abstract <p>It was earlier shown that the product <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(xy\)</EquationSource> <!--CMatCMGU2570019Voronenko-m1--> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k=6l\pm 1\)</EquationSource> <!--CMatCMGU2570019Voronenko-m2--> </InlineEquation> is a universal function in the class of linear functions of two variables. Subsequently, the existence of universal polynomials was proved in the class of linear functions of any number of variables for arbitrary <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k\)</EquationSource> <!--CMatCMGU2570019Voronenko-m3--> </InlineEquation>. This study proves that the polynomial <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(xy\)</EquationSource> <!--CMatCMGU2570019Voronenko-m4--> </InlineEquation> is universal in the class of linear functions over the Galois field <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(GF(p^{m})\)</EquationSource> <!--CMatCMGU2570019Voronenko-m5--> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(p\)</EquationSource> <!--CMatCMGU2570019Voronenko-m6--> </InlineEquation> is a prime number and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(m\)</EquationSource> <!--CMatCMGU2570019Voronenko-m7--> </InlineEquation> is a natural number, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(m\geqslant 2\)</EquationSource> <!--CMatCMGU2570019Voronenko-m8--> </InlineEquation>.</p>

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Universality of the Product for Classes of Linear Functions over a Galois Field

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • A. A. Voronenko,
  • E. A. Selyankina

摘要

Abstract

It was earlier shown that the product \(xy\) for \(k=6l\pm 1\) is a universal function in the class of linear functions of two variables. Subsequently, the existence of universal polynomials was proved in the class of linear functions of any number of variables for arbitrary \(k\) . This study proves that the polynomial \(xy\) is universal in the class of linear functions over the Galois field \(GF(p^{m})\) , where \(p\) is a prime number and \(m\) is a natural number, \(m\geqslant 2\) .