Symmetric polynomial

In mathematics , a polynomial is called symmetrical in several indeterminates , if the indeterminates can be interchanged without changing the polynomial.

Formal definition

Let it be a natural number , a ring . Then a polynomial is called symmetric in if ${\displaystyle n>1}$ ${\displaystyle A}$ ${\displaystyle p\in A[X_{1},\ldots ,X_{n}]}$ ${\displaystyle X_{1},\ldots ,X_{n}}$

${\displaystyle p(X_{\sigma (1)},\ldots ,X_{\sigma (n)})=p(X_{1},\ldots ,X_{n})}$for all permutations ${\displaystyle \sigma \in S_{n}}$

applies.

Equivalent descriptions are:

• For everyone is ${\displaystyle k\neq m}$

${\displaystyle p(X_{1},\ldots ,X_{k-1},X_{k},X_{k+1}\ldots ,X_{m-1},X_{m},X_{m+1},\ldots ,X_{n})=p(X_{1},\ldots ,X_{k-1},X_{m},X_{k+1},\ldots ,X_{m-1},X_{k},X_{m+1},\ldots ,X_{n}),}$

that is, one can exchange any two indeterminates for one another.

• Be it

${\displaystyle p=\sum _{e_{1}\geq 0,\ldots ,e_{n}\geq 0}a_{e_{1},\ldots ,e_{n}}X_{1}^{e_{1}}\cdots X_{n}^{e_{n}}.}$

Then is symmetric if and only if ${\displaystyle p}$

${\displaystyle a_{e_{1},\ldots ,e_{n}}=a_{e_{\sigma (1)},\ldots ,e_{\sigma (n)}}}$ for all ${\displaystyle \sigma \in S_{n}}$

applies. Clearly this means that the coefficient of a monomial of just depends on which exponent how often happen and not in which indeterminate. ${\displaystyle p}$

• The symmetrical group operates through ${\displaystyle S_{n}}$

${\displaystyle (\sigma p)(X_{1},\ldots ,X_{n})=p(X_{\sigma (1)},\ldots ,X_{\sigma (n)})}$

on the polynomial ring . A polynomial is symmetric if and only if it is invariant under this operation, i.e. i.e. if ${\displaystyle A[X_{1},\ldots ,X_{n}]}$

${\displaystyle \sigma p=p}$ for all ${\displaystyle \sigma \in S_{n}}$

applies. One possible notation for the ring of symmetrical polynomials is therefore

${\displaystyle A[X_{1},\ldots ,X_{n}]^{S_{n}}.}$

Body of symmetrical functions

We are now replacing the base ring with a base body . The body of symmetric functions is analogous to the above definition of the fixed field below , thus: . The expansion of the body is Galois with the Galois group and thus has degrees ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle S_{n}}$ ${\displaystyle L=K(X_{1},\ldots ,X_{n})^{S_{n}}}$
${\displaystyle K(X_{1},\ldots ,X_{n})/L}$ ${\displaystyle S_{n}}$ ${\displaystyle n!}$

Examples

• The polynomial is symmetric in and , but not symmetric in . ${\displaystyle X+Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X,Y,Z}$
• A symmetrical polynomial can be formed from any arbitrary polynomial in the variables by adding the images under the permutations, i.e.: ${\displaystyle P}$ ${\displaystyle X_{1},\ldots ,X_{n}}$

${\displaystyle \sum \limits _{\sigma \in S_{n}}\sigma (P)}$

Elementary symmetric polynomials

A particularly important type of symmetric polynomials are the so-called elementary symmetric polynomials. They are the basic building blocks of symmetrical polynomials in the sense that the latter can always be expressed as a polynomial in the former, and this in only one way.

For every number ( degree of symmetry) of indeterminates and every (polynomial) degree there is exactly one elementary symmetric polynomial . ${\displaystyle n}$ ${\displaystyle k\leq n}$ ${\displaystyle \sigma _{n,k}}$

→ Main article : Elementary symmetric polynomial

Examples

• The two elementary symmetric polynomials in the variables , are ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle \sigma _{2,1}=X+Y\qquad }$ such as ${\displaystyle \sigma _{2,2}=X\cdot Y}$

• In three variables , , one has the three elementary symmetric polynomials ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$

${\displaystyle \sigma _{3,1}=X+Y+Z\qquad \sigma _{3,2}=X\cdot Y+X\cdot Z+Y\cdot Z\qquad \sigma _{3,3}=X\cdot Y\cdot Z}$

Power sums

With the power sums

${\displaystyle s_{n,m}(X_{1},\ldots ,X_{n}):=X_{1}^{m}+\ldots +X_{n}^{m}}$, ${\displaystyle m=0,1,2,\ldots }$

for one has another kind of symmetric polynomial. They are connected to the elementary symmetric polynomials via the Newton identities . For example, you have: ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle \sigma _{n,k}}$ ${\displaystyle m=1,2,3}$

• ${\displaystyle s_{n,1}=X_{1}+\dotsb +X_{n}=\sigma _{n,1}}$
• ${\displaystyle s_{n,2}=X_{1}^{2}+\dotsb +X_{n}^{2}=\sigma _{n,1}^{2}-2\sigma _{n,2}}$
• ${\displaystyle s_{n,3}=X_{1}^{3}+\dotsb +X_{n}^{3}=\sigma _{n,1}^{3}-3\sigma _{n,1}\sigma _{n,2}+3\sigma _{n,3}}$

And vice versa:

• ${\displaystyle \sigma _{n,1}=s_{n,1}}$
• ${\displaystyle 2\sigma _{n,2}=s_{n,1}^{2}-s_{n,2}}$
• ${\displaystyle 6\sigma _{n,3}=s_{n,1}^{3}-3s_{n,1}s_{n,2}+2s_{n,3}}$

If the ring contains the rational numbers , a similar theorem applies as for the elementary symmetric polynomials: ${\displaystyle A}$ ${\displaystyle \mathbb {Q} }$

• Every symmetric polynomial can be written as a polynomial in power sums.
• This representation is clear .

See also

• Lagrange resolvent
• Symmetrical function

literature

• Siegfried Bosch : Algebra. 8th edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-642-39566-6 , Chapter 4, Section 4.
• Gerd Fischer : Textbook of Algebra. 3. Edition. Springer, Wiesbaden 2013, ISBN 978-3-658-02220-4 , Chapter III, §4.1.
• Jens Carsten Jantzen , Joachim Schwermer : Algebra. 2nd Edition. Springer, Berlin / Heidelberg 2014, ISBN 978-3-642-40532-7 , Chapter IV, §3.3.