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SymmetricReduction
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SymmetricReduction

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SymmetricReduction[f,{x1,…,xn}]

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

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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:

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-hxv04f
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-mp9cun
Out[1]=1

Name the first two elementary symmetric polynomials s1 and s2:

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-3yacv
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-d3ls52
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8bx0ahua-ko6jp6
Out[2]=2

SymmetricReduction will reduce the polynomial part of an expression:

In[1]:=1
✖
https://wolfram.com/xid/0b8bx0ahua-yj7
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 , , :

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-b2n
In[2]:=2
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https://wolfram.com/xid/0b8bx0ahua-png
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b8bx0ahua-beb
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0b8bx0ahua-fpi
Out[4]=4

Use SymmetricReduction to solve for , , :

In[5]:=5
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https://wolfram.com/xid/0b8bx0ahua-k2f
Out[5]=5

The monic polynomial with roots , , :

In[6]:=6
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https://wolfram.com/xid/0b8bx0ahua-usb
Out[6]=6

Check:

In[7]:=7
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https://wolfram.com/xid/0b8bx0ahua-tux
Out[7]=7
In[8]:=8
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https://wolfram.com/xid/0b8bx0ahua-v68
Out[8]=8

Consider solving the following symmetric system of equations:

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-map

Use ChebyshevT to convert to a symmetric system of polynomials:

In[2]:=2
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https://wolfram.com/xid/0b8bx0ahua-bp4
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0b8bx0ahua-cyv
In[4]:=4
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https://wolfram.com/xid/0b8bx0ahua-yd7
Out[4]=4

The leaf count of the solution is enormous:

In[5]:=5
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https://wolfram.com/xid/0b8bx0ahua-vmx
Out[5]=5

Convert to a system of equations of symmetric polynomials :

In[6]:=6
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https://wolfram.com/xid/0b8bx0ahua-dtu
Out[6]=6

Solve the new system of equations:

In[7]:=7
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https://wolfram.com/xid/0b8bx0ahua-bdb
Out[7]=7

The leaf count of the symmetric solution is much smaller:

In[8]:=8
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https://wolfram.com/xid/0b8bx0ahua-p5y
Out[8]=8

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

In[9]:=9
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https://wolfram.com/xid/0b8bx0ahua-kt5
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:

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-veia4
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8bx0ahua-ckij7b
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:

In[1]:=1
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https://wolfram.com/xid/0b8bx0ahua-irs
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0b8bx0ahua-dqz
Out[2]=2

Check:

In[3]:=3
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https://wolfram.com/xid/0b8bx0ahua-gra
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0b8bx0ahua-op8
Out[4]=4

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

In[5]:=5
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https://wolfram.com/xid/0b8bx0ahua-eux
Out[5]=5
Wolfram Research (2007), SymmetricReduction, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetricReduction.html.
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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.

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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.

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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

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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 ]}

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@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 ]}

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@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 ]}