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Abstract <p> A homogeneous polynomial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S(x_1, \ldots, x_n)\)</EquationSource> </InlineEquation> of degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(r\)</EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation> variables possesses a discriminant <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(D_{n|r}(S)\)</EquationSource> </InlineEquation>, which vanishes if and only if the system of equations <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\partial S / \partial x_i = 0\)</EquationSource> </InlineEquation> has nontrivial solutions. We provide an explicit formula for the discriminants of symmetric (under permutations of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(x_1, \ldots, x_n\)</EquationSource> </InlineEquation>) homogeneous polynomials of degree <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(r\)</EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(n \geq r\)</EquationSource> </InlineEquation> variables. This formula is highly effective from a computational perspective: symbolic computer calculations using this formula take seconds even for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(n \approx 20\)</EquationSource> </InlineEquation>. We work out the cases <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(r = 2\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(3\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(4\)</EquationSource> </InlineEquation> in detail. We also consider the case of completely antisymmetric polynomials. </p>

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Discriminants of symmetric polynomials

  • 期刊论文

  • 发表日期：

    2026年5月26日

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  • N. S. Perminov,
  • S. R. Shakirov

摘要

Abstract

A homogeneous polynomial \(S(x_1, \ldots, x_n)\) of degree \(r\) in \(n\) variables possesses a discriminant \(D_{n|r}(S)\) , which vanishes if and only if the system of equations \(\partial S / \partial x_i = 0\) has nontrivial solutions. We provide an explicit formula for the discriminants of symmetric (under permutations of \(x_1, \ldots, x_n\) ) homogeneous polynomials of degree \(r\) in \(n \geq r\) variables. This formula is highly effective from a computational perspective: symbolic computer calculations using this formula take seconds even for \(n \approx 20\) . We work out the cases \(r = 2\) , \(3\) , and \(4\) in detail. We also consider the case of completely antisymmetric polynomials.