# SymmetricReduction

SymmetricReduction[f,{x1,,xn}]

gives a pair of polynomials in such that , where is the symmetric part and is the remainder.

SymmetricReduction[f,{x1,,xn},{s1,,sn}]

gives the pair with the elementary symmetric polynomials in replaced by .

# Details • 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 .

# Examples

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## 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:

## Scope(2)

SymmetricReduction will reduce the polynomial part of an expression:

## Applications(2)

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 x1,x2,x3:

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 x1,x2,x3 in terms of the symmetric polynomials is also quick:

## Properties & Relations(2)

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: