BUILT-IN MATHEMATICA SYMBOL

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

.

MORE INFORMATION

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

and

:

Write a symmetric polynomial as a sum of elementary symmetric polynomials:

In[1]:=

Out[1]=

Write a nonsymmetric polynomial as a symmetric part and a remainder:

In[1]:=

Out[1]=

Name the first two elementary symmetric polynomials

and

:

In[1]:=

Out[1]=

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

:

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:

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-x_it, {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: