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SymmetricReduction

SymmetricReduction[f, {x1, ..., xn}]
gives a pair of polynomials {p,q} in x_1,...,x_n such that f==p+q, where p is the symmetric part and q is the remainder.
SymmetricReduction[f, {x1, ..., xn}, {s1, ..., sn}]
gives the pair {p,q} with the elementary symmetric polynomials in p replaced by s_1,...,s_n.
  • If f is a symmetric polynomial, then p is the unique polynomial in elementary symmetric polynomials equal to f, and q is zero.
  • If f is not a symmetric polynomial, then the output p is not unique, but depends on the ordering of its variables.
  • For a given ordering, a nonsymmetric polynomial f can be expressed uniquely as a sum of its symmetric part p and a remainder q that does not contain descending monomials. A monomial is called descending if e_1>=...>=e_n.
  • Changing the ordering of the variables may produce different pairs {p,q}.
  • SymmetricReduction does not check to see that f is a polynomial, and will attempt to symmetrize the polynomial part of f.
EXAMPLESCLOSE ALL
Basic Examples   (3)
Write a symmetric polynomial as a sum of elementary symmetric polynomials:
Write a nonsymmetric polynomial as a symmetric part and a remainder:
Name the first two elementary symmetric polynomials s1 and s2:
Write a symmetric polynomial as a sum of elementary symmetric polynomials:
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Write a nonsymmetric polynomial as a symmetric part and a remainder:
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Name the first two elementary symmetric polynomials s1 and s2:
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Scope   (2)
SymmetricReduction will reduce the polynomial part of an expression:
Applications   (2)
Let the roots of the equation x^3+a x^2+b x+c==0 be Alpha, Beta, Gamma. The coefficients a, b, c are trivially related to the symmetric polynomials of Alpha, Beta, Gamma:
A similar expression holds for the monic polynomial with roots alpha^2, beta^2, gamma^2:
Use SymmetricReduction to solve for , , :
The monic polynomial with roots alpha^2, beta^2, gamma^2:
Check:
Consider solving the following symmetric system of equations:
Use ChebyshevT to convert to a symmetric system of polynomials:
Solve is able to solve the equations in the variables x1, x2, x3:
The leaf count of the solution is enormous:
Convert to a system of equations of symmetric polynomials s1, s2, s3:
Solve the new system of equations:
The leaf count of the symmetric solution is much smaller:
Solving for the variables x_1,x_2,x_3 in terms of the symmetric polynomials s_1,s_2,s_3 is also quick:
Properties & Relations   (2)
The order of variables can effect the decomposition into symmetric and nonsymmetric parts:
Another basis for the symmetric polynomials consists of the complete symmetric polynomials. They are the sum of all monomials of a given degree, and can be defined by the generating function Product[1-xit, {i, n}]-1:
A determinant formula expresses the elementary symmetric polynomials in the basis of the complete symmetric polynomials:
Check:
Any symmetric polynomial can also be expressed in terms of the complete symmetric polynomials:
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