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symmetric polynomial
en
(algebra) A polynomial which remains unchanged if its variables are permuted
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In analogy to the derivation for curves in 3.11, we use the symmetric polynomial b[XI ... xn] of b(x).
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The ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity.
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This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions).
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The remaining n elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only.
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The Pfaffian is an invariant polynomial of a skew-symmetric matrix under a proper orthogonal change of basis.
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The Pfaffian (considered as a polynomial) is nonvanishing only for 2n × 2n skew-symmetric matrices, in which case it is a polynomial of degree n.
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Special kinds of symmetric polynomials[edit]
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The value of this polynomial, when applied to the coefficients of a skew-symmetric matrix, is called the Pfaffian of that matrix.
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In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the size of the matrix.
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If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two nonreal roots, then the Galois group of f is the full symmetric group Sp.
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Some polynomials are not symmetrical.
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These interactions give rise to several identities, often expressed in terms of generating functions or symmetric functions (Schur functions, Macdonald polynomials, etc.).
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Minkowski's theorem relates the number d(Λ) and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial.
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For the porcelain tile with appropriate cutting behaviour, the residual stress profile in the body was symmetrical and could be fitted by just using the second- degree Legendre polynomial.
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We define the symmetric Rad ́on-Nikod ́ym property for s-tensor norms and show, under certain hypothesis, that maximal polynomial ideals associated with norms with this property coincide isometrically with their minimal kernel.
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