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Abstract <p>In this paper we formulate and prove an equality expressing the discriminant of a sum of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--VestSPGU2570046Bekker-m1--> </InlineEquation> degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d\)</EquationSource> <!--VestSPGU2570046Bekker-m2--> </InlineEquation> polynomials in one variable, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n \geqslant 2d - 1\)</EquationSource> <!--VestSPGU2570046Bekker-m3--> </InlineEquation>, in terms of subsums of those polynomials. To that end we study the properties of polarizations of arbitrary degree of maps <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(f:M \to N\)</EquationSource> <!--VestSPGU2570046Bekker-m4--> </InlineEquation> of abelian groups, defined as certain generalizations of the classical polar form of a homogeneous polynomial in several variables. We show that the vanishing of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((n + 1)\)</EquationSource> <!--VestSPGU2570046Bekker-m5--> </InlineEquation>-st polarization of a map <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(f\)</EquationSource> <!--VestSPGU2570046Bekker-m6--> </InlineEquation> is equivalent to polylinearity of its <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--VestSPGU2570046Bekker-m7--> </InlineEquation>th polarization. In case <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(M = {R^s}\)</EquationSource> <!--VestSPGU2570046Bekker-m8--> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(N = R\)</EquationSource> <!--VestSPGU2570046Bekker-m9--> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(f\)</EquationSource> <!--VestSPGU2570046Bekker-m10--> </InlineEquation> is given by a polynomial of degree <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--VestSPGU2570046Bekker-m11--> </InlineEquation> we prove that its <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\((n + 1)\)</EquationSource> <!--VestSPGU2570046Bekker-m12--> </InlineEquation>-st polarization vanishes and simultaneously obtain a formula for its <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--VestSPGU2570046Bekker-m13--> </InlineEquation>th&#xa0;polarization that recovers the classical polarization identity.</p>

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Polarization Identities

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • B. M. Bekker,
  • O. B. Podkopaev

摘要

Abstract

In this paper we formulate and prove an equality expressing the discriminant of a sum of \(n\) degree \(d\) polynomials in one variable, where \(n \geqslant 2d - 1\) , in terms of subsums of those polynomials. To that end we study the properties of polarizations of arbitrary degree of maps \(f:M \to N\) of abelian groups, defined as certain generalizations of the classical polar form of a homogeneous polynomial in several variables. We show that the vanishing of \((n + 1)\) -st polarization of a map \(f\) is equivalent to polylinearity of its \(n\) th polarization. In case \(M = {R^s}\) , \(N = R\) and \(f\) is given by a polynomial of degree \(n\) we prove that its \((n + 1)\) -st polarization vanishes and simultaneously obtain a formula for its \(n\) th polarization that recovers the classical polarization identity.