SymmetricReduction
✖
SymmetricReduction
gives a pair of polynomials in
such that
, where
is the symmetric part and
is the remainder.
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
open allclose allBasic Examples (3)Summary of the most common use cases
Write a symmetric polynomial as a sum of elementary symmetric polynomials:

https://wolfram.com/xid/0b8bx0ahua-hxv04f

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

https://wolfram.com/xid/0b8bx0ahua-mp9cun

Name the first two elementary symmetric polynomials s1 and s2:

https://wolfram.com/xid/0b8bx0ahua-3yacv

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

https://wolfram.com/xid/0b8bx0ahua-d3ls52


https://wolfram.com/xid/0b8bx0ahua-ko6jp6

SymmetricReduction will reduce the polynomial part of an expression:

https://wolfram.com/xid/0b8bx0ahua-yj7

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
,
,
:

https://wolfram.com/xid/0b8bx0ahua-b2n

https://wolfram.com/xid/0b8bx0ahua-png


https://wolfram.com/xid/0b8bx0ahua-beb

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

https://wolfram.com/xid/0b8bx0ahua-fpi

Use SymmetricReduction to solve for ,
,
:

https://wolfram.com/xid/0b8bx0ahua-k2f

The monic polynomial with roots ,
,
:

https://wolfram.com/xid/0b8bx0ahua-usb


https://wolfram.com/xid/0b8bx0ahua-tux


https://wolfram.com/xid/0b8bx0ahua-v68

Consider solving the following symmetric system of equations:

https://wolfram.com/xid/0b8bx0ahua-map
Use ChebyshevT to convert to a symmetric system of polynomials:

https://wolfram.com/xid/0b8bx0ahua-bp4

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

https://wolfram.com/xid/0b8bx0ahua-cyv

https://wolfram.com/xid/0b8bx0ahua-yd7

The leaf count of the solution is enormous:

https://wolfram.com/xid/0b8bx0ahua-vmx

Convert to a system of equations of symmetric polynomials :

https://wolfram.com/xid/0b8bx0ahua-dtu

Solve the new system of equations:

https://wolfram.com/xid/0b8bx0ahua-bdb

The leaf count of the symmetric solution is much smaller:

https://wolfram.com/xid/0b8bx0ahua-p5y

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

https://wolfram.com/xid/0b8bx0ahua-kt5

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:

https://wolfram.com/xid/0b8bx0ahua-veia4


https://wolfram.com/xid/0b8bx0ahua-ckij7b

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:

https://wolfram.com/xid/0b8bx0ahua-irs

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

https://wolfram.com/xid/0b8bx0ahua-dqz


https://wolfram.com/xid/0b8bx0ahua-gra


https://wolfram.com/xid/0b8bx0ahua-op8

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

https://wolfram.com/xid/0b8bx0ahua-eux

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
]}
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
]}