Elementary symmetric polynomial

In mathematics , especially in commutative algebra , the elementary symmetric polynomials are basic building blocks of symmetric 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. ${\displaystyle n}$ ${\displaystyle k\leq n}$

definition

They are indefinite. The coefficients of ${\displaystyle T,X_{1},\ldots ,X_{n}}$

${\displaystyle (T+X_{1})(T+X_{2})\dotsm (T+X_{n})=T^{n}+\sigma _{1}T^{n-1}+\sigma _{2}T^{n-2}+\dotsb +\sigma _{n}}$

as a polynomial in are symmetric in ; they are called elementary symmetric polynomials . They can be specified explicitly as ${\displaystyle T}$ ${\displaystyle X_{1},\dotsc ,X_{n}}$

${\displaystyle \sigma _{1}=X_{1}+\dotsb +X_{n}}$

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

${\displaystyle \ldots }$

${\displaystyle \sigma _{k}=\sum _{1\leq i_{1}<i_{2}<\dotsb <i_{k}\leq n}X_{i_{1}}\dotsm X_{i_{k}}}$

${\displaystyle \ldots }$

${\displaystyle \sigma _{n}=X_{1}\dotsm X_{n}}$

You can also write as ${\displaystyle \sigma _{k}}$

${\displaystyle \sigma _{k}=\sum _{S\subseteq \{1,\dotsc ,n\} \atop \#S=k}\ \prod _{i\in S}X_{i}\ .}$

Examples

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

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

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

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

properties

• In an elementary symmetric polynomial, the monomials have a uniform degree: it is a homogeneous polynomial .
• If you add the degree of as the first index, then is for ${\displaystyle n}$ ${\displaystyle S_{n}}$ ${\displaystyle n=2}$

	${\displaystyle \sigma _{2,1}(X_{1},X_{2})}$	${\displaystyle =}$	${\displaystyle X_{1}}$	${\displaystyle +}$	${\displaystyle X_{2}}$
	${\displaystyle \sigma _{2,2}(X_{1},X_{2})}$	${\displaystyle =}$			${\displaystyle X_{1}\cdot X_{2}}$
The elementary symmetric polynomials for can be calculated recursively as follows: ${\displaystyle n>2}$
	${\displaystyle \sigma _{n,1}(X_{1},\ldots ,X_{n})}$	${\displaystyle =}$	${\displaystyle \sigma _{n-1,1}(X_{1},\ldots ,X_{n-1})}$	${\displaystyle +}$	${\displaystyle X_{n}}$
	${\displaystyle \sigma _{n,k}(X_{1},\ldots ,X_{n})}$	${\displaystyle =}$	${\displaystyle \sigma _{n-1,k}(X_{1},\ldots ,X_{n-1})}$	${\displaystyle +}$	${\displaystyle \sigma _{n-1,k-1}(X_{1},\ldots ,X_{n-1})\cdot X_{n}}$	${\displaystyle (k\in \{2,\dotsc ,n-1\})}$
	${\displaystyle \sigma _{n,n}(X_{1},\ldots ,X_{n})}$	${\displaystyle =}$			${\displaystyle \sigma _{n-1,n-1}(X_{1},\ldots ,X_{n-1})\cdot X_{n}}$

• The elementary symmetric polynomial of degree of symmetry and degree of polynomial contains monomials. ${\displaystyle \sigma _{n,k}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k\in \{1,\dotsc ,n\}}$ ${\displaystyle {\binom {n}{k}}}$

• Law of the elementary symmetric polynomials:

${\displaystyle A[X_{1},\dotsc ,X_{n}]^{S_{n}}=A[\sigma _{1},\dotsc ,\sigma _{n}]}$

In words: Every symmetrical polynomial can be written as a polynomial in the elementary symmetrical polynomials. The sentence comes from Joseph-Louis Lagrange , but was already known to Isaac Newton .

More precisely, it is even true that this representation is unambiguous , because:

• The elementary symmetric polynomials are algebraically independent .

That means: There are no two different polynomials in the variables for which the following applies: ${\displaystyle P_{1},\,P_{2}}$ ${\displaystyle X_{1},\dotsc ,X_{n}}$ ${\displaystyle P_{1}(\sigma _{1},\dotsc ,\sigma _{n})=P_{2}(\sigma _{1},\dotsc ,\sigma _{n})}$

• There are a integral domain , ${\displaystyle A}$

${\displaystyle p(T)=T^{n}+a_{1}T^{n-1}+a_{2}T^{n-2}+\dotsb +a_{n}}$

a polynomial with coefficients in and the zeros (counted with multiplicity) of in an algebraic closure of the quotient field of . Then according to the law of roots of Vieta : ${\displaystyle A}$ ${\displaystyle x_{1},\dotsc ,x_{n}}$ ${\displaystyle p}$ ${\displaystyle A}$

${\displaystyle a_{1}=-(x_{1}+\dotsb +x_{n})}$

${\displaystyle a_{2}=x_{1}x_{2}+x_{1}x_{3}+\dotsb +x_{1}x_{n}+x_{2}x_{3}+\dotsb +x_{n-1}x_{n}}$

${\displaystyle \ldots }$

${\displaystyle a_{k}=(-1)^{k}\cdot \sigma _{k}(x_{1},\dotsc ,x_{n})}$

${\displaystyle \ldots }$

${\displaystyle a_{n}=(-1)^{n}x_{1}\dotsm x_{n}.}$

calculation

With numerical values ​​(instead of indeterminate), the calculation is particularly simple - instead of monomials consisting of products with up to factors, you only have multiplications. ${\displaystyle 2^{n}}$ ${\displaystyle n}$ ${\displaystyle {\binom {n}{2}}}$

With the following program the coefficients of the polynomial ${\displaystyle s_{k}}$

${\displaystyle p(T)=T^{n}+\sum _{k=1}^{n}(-1)^{k}s_{k}\,T^{n-k}}$

from the roots of the polynomial ${\displaystyle x_{k}}$

${\displaystyle p(T)=\prod _{k=1}^{n}(T-x_{k})}$

to calculate:

// Umwandlung von Nullstellen in Koeffizienten: double x[]; // bei Eingabe: n Zahlen für die Nullstellen x[1, ... ,n] // bei Ausgabe: n Zahlen für die Koeffizienten s[1, ... ,n] for (m=2; m≤n; ++m) { // leere Schleife, wenn n ≤ 1 y = x[m]; x[m] *= x[m-1]; // ${\displaystyle \sigma _{m,m}(x_{1},...,x_{m})=\sigma _{m-1,m-1}(x_{1},...,x_{m-1})\,x_{m}}$ for (k=m-1; k≥2; --k) { // leere Schleife, wenn m ≤ 2 x[k] += x[k-1]*y; // ${\displaystyle \sigma _{m,k}(x_{1},...,x_{m})=\sigma _{m-1,k}(x_{1},...,x_{m-1})\,+\,\sigma _{m-1,k-1}(x_{1},...,x_{m-1})\,x_{m}}$ } x[1] += y; // ${\displaystyle \sigma _{m,1}(x_{1},...,x_{m})=\sigma _{m-1,1}(x_{1},...,x_{m-1})\,+\,x_{m}}$ }

Examples

• ${\displaystyle X_{1}^{2}+\dotsb +X_{n}^{2}=\sigma _{1}^{2}-2\sigma _{2}}$
• ${\displaystyle X_{1}^{3}+\dotsb +X_{n}^{3}=\sigma _{1}^{3}-3\sigma _{1}\sigma _{2}+3\sigma _{3}.}$In general, the power sums are connected to the elementary symmetric polynomials by the Newton identities .
• The polynomial

${\displaystyle \Delta (X_{1},\dotsc ,X_{n})=\prod _{i<j}(X_{i}-X_{j})^{2}=(-1)^{n(n-1)/2}\prod _{i\neq j}(X_{i}-X_{j})}$

is symmetric in , so it can be written as a polynomial in the elementary symmetric polynomials. Is now ${\displaystyle X_{1},\dotsc ,X_{n}}$

${\displaystyle p(T)=T^{n}+a_{1}T^{n-1}+a_{2}T^{n-2}+\dotsb +a_{n}}$

a polynomial with zeros as above and if these are inserted into , then the elementary symmetric expressions correspond to the coefficients except for the signs , i.e. i.e., is a polynomial in the coefficients that only depends on . With the exception of definition variants for the sign, this polynomial is the discriminant of . ${\displaystyle x_{1},\dotsc ,x_{n}}$ ${\displaystyle \Delta }$ ${\displaystyle a_{i}}$ ${\displaystyle \Delta (x_{1},\dotsc ,x_{n})}$ ${\displaystyle n}$ ${\displaystyle a_{1},\dotsc ,a_{n}}$ ${\displaystyle p}$

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.

References and comments

1. ↑ Jantzen, Schwermer: Algebra 2014, Chapter IV, Theorem 3.5.