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SymmetricReduction

SymmetricReduction
gives a pair of polynomials in such that , where is the symmetric part and is the remainder.
SymmetricReduction
gives the pair with the elementary symmetric polynomials in replaced by .
  • If is a symmetric polynomial, then is the unique polynomial in elementary symmetric polynomials equal to , and is zero.
  • If is not a symmetric polynomial, then the output is not unique, but depends on the ordering of its variables.
  • For a given ordering, a nonsymmetric polynomial can be expressed uniquely as a sum of its symmetric part and a remainder that does not contain descending monomials. A monomial is called descending if .
  • Changing the ordering of the variables may produce different pairs .
  • SymmetricReduction does not check to see that is a polynomial, and will attempt to symmetrize the polynomial part of .
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 and :
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 and :
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SymmetricReduction will reduce the polynomial part of an expression:
Let the roots of the equation be , , . The coefficients , , are trivially related to the symmetric polynomials of , , :
A similar expression holds for the monic polynomial with roots , , :
Use SymmetricReduction to solve for , , :
The monic polynomial with roots , , :
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 :
The leaf count of the solution is enormous:
Convert to a system of equations of symmetric polynomials :
Solve the new system of equations:
The leaf count of the symmetric solution is much smaller:
Solving for the variables in terms of the symmetric polynomials is also quick:
The order of variables can affect 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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