WOLFRAM

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 f is a symmetric polynomial, then is the unique polynomial in elementary symmetric polynomials equal to f, and is zero.
  • If f 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 f 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 f is a polynomial, and will attempt to symmetrize the polynomial part of f.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

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

Out[1]=1

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

Out[1]=1

Name the first two elementary symmetric polynomials s1 and s2:

Out[1]=1

Scope  (2)Survey of the scope of standard use cases

Out[1]=1
Out[2]=2

SymmetricReduction will reduce the polynomial part of an expression:

Out[1]=1

Applications  (2)Sample problems that can be solved with this function

Let the roots of the equation be , , . The coefficients , , are trivially related to the symmetric polynomials of , , :

Out[2]=2
Out[3]=3

A similar expression holds for the monic polynomial with roots , , :

Out[4]=4

Use SymmetricReduction to solve for , , :

Out[5]=5

The monic polynomial with roots , , :

Out[6]=6

Check:

Out[7]=7
Out[8]=8

Consider solving the following symmetric system of equations:

Use ChebyshevT to convert to a symmetric system of polynomials:

Out[2]=2

Solve is able to solve the equations in the variables x1,x2,x3:

Out[4]=4

The leaf count of the solution is enormous:

Out[5]=5

Convert to a system of equations of symmetric polynomials :

Out[6]=6

Solve the new system of equations:

Out[7]=7

The leaf count of the symmetric solution is much smaller:

Out[8]=8

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

Out[9]=9

Properties & Relations  (2)Properties of the function, and connections to other functions

The order of variables can affect the decomposition into symmetric and nonsymmetric parts:

Out[1]=1
Out[2]=2

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:

Out[1]=1

A determinant formula expresses the elementary symmetric polynomials in the basis of the complete symmetric polynomials:

Out[2]=2

Check:

Out[3]=3
Out[4]=4

Any symmetric polynomial can also be expressed in terms of the complete symmetric polynomials:

Out[5]=5
Wolfram Research (2007), SymmetricReduction, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetricReduction.html.
Wolfram Research (2007), SymmetricReduction, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetricReduction.html.

Text

Wolfram Research (2007), SymmetricReduction, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetricReduction.html.

Wolfram Research (2007), SymmetricReduction, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetricReduction.html.

CMS

Wolfram Language. 2007. "SymmetricReduction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SymmetricReduction.html.

Wolfram Language. 2007. "SymmetricReduction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SymmetricReduction.html.

APA

Wolfram Language. (2007). SymmetricReduction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SymmetricReduction.html

Wolfram Language. (2007). SymmetricReduction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SymmetricReduction.html

BibTeX

@misc{reference.wolfram_2025_symmetricreduction, author="Wolfram Research", title="{SymmetricReduction}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetricReduction.html}", note=[Accessed: 08-July-2025 ]}

@misc{reference.wolfram_2025_symmetricreduction, author="Wolfram Research", title="{SymmetricReduction}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetricReduction.html}", note=[Accessed: 08-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_symmetricreduction, organization={Wolfram Research}, title={SymmetricReduction}, year={2007}, url={https://reference.wolfram.com/language/ref/SymmetricReduction.html}, note=[Accessed: 08-July-2025 ]}

@online{reference.wolfram_2025_symmetricreduction, organization={Wolfram Research}, title={SymmetricReduction}, year={2007}, url={https://reference.wolfram.com/language/ref/SymmetricReduction.html}, note=[Accessed: 08-July-2025 ]}