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Difference between revisions of "Elementary symmetric sum"

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== Definition ==
 
== Definition ==
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The <math>k</math>-th '''elmentary symmetric sum''' of a [[set]] of <math>n</math> numbers is the sum of all products of <math>k</math> of those numbers (<math>1 \leq k \leq n</math>).  For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
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The <math>k</math>-th '''elementary symmetric sum''' of a [[set]] of <math>n</math> numbers is the sum of all products of <math>k</math> of those numbers (<math>1 \leq k \leq n</math>).  For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
  
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1st Symmetric Sum = <math>e_1 = a+b+c+d</math>
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1st Symmetric Sum = <math>S_1 = a+b+c+d</math>
  
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2nd Symmetric Sum = <math>e_2 = ab+ac+ad+bc+bd+cd</math>
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2nd Symmetric Sum = <math>S_2 = ab+ac+ad+bc+bd+cd</math>
  
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3rd Symmetric Sum = <math>e_3 = abc+abd+acd+bcd</math>
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3rd Symmetric Sum = <math>S_3 = abc+abd+acd+bcd</math>
  
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4th Symmetric Sum = <math>e_4 = abcd</math>
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4th Symmetric Sum = <math>S_4 = abcd</math>
  
 
==Notation==
 
==Notation==
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The first elmentary symmetric sum of <math>f(x)</math> is often written <math>\sum_{sym}f(x)</math>. The <math>n</math>th can be written <math>\sum_{sym}^{n}f(x)</math>
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The first elementary symmetric sum of <math>f(x)</math> is often written <math>\sum_{sym}f(x)</math>. The <math>n</math>th can be written <math>\sum_{sym}^{n}f(x)</math>
  
 
== Uses ==
 
== Uses ==
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Any symmetric sum can be written as a [[polynomial]] of the elmentary symmetric sum functions. For example, <math>x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = e_1^3 - 3e_1e_2 + 3e_3</math>. This is often used to solve systems of equations involving [[power sum]]s, combined with Vieta's.
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Any [[symmetric sum]] can be written as a [[polynomial]] of the elementary symmetric sum functions. For example, <math>x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = S_1^3 - 3S_1S_2 + 3S_3</math>. This is often used to solve systems of equations involving [https://en.wikipedia.org/wiki/Sums_of_powers sums of powers], combined with Vieta's formulas.
  
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Elmentary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial, the coefficient of the <math>x^1</math> term is <math>e_1</math>, and the coefficient of the <math>x^k</math> term is <math>e_k</math>.  
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Elementary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial of degree <math>n</math>, the coefficient of the <math>x^n</math> term is <math>(-1)^nS_n</math>, and the coefficient of the <math>x^k</math> term is <math>(-1)^{n-k}S_{n-k}</math>, where the symmetric sums are taken over the roots of the polynomial.
  
 
==See Also==
 
==See Also==
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*[[Cyclic sum]]
 
*[[Cyclic sum]]
  
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[[Category:Elementary algebra]]
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[[Category:Algebra]]
 
[[Category:Definition]]
 
[[Category:Definition]]

Latest revision as of 11:51, 26 July 2023

An elementary symmetric sum is a type of summation.

Definition

The $k$-th elementary symmetric sum of a set of $n$ numbers is the sum of all products of $k$ of those numbers ($1 \leq k \leq n$). For example, if $n = 4$, and our set of numbers is $\{a, b, c, d\}$, then:

1st Symmetric Sum = $S_1 = a+b+c+d$

2nd Symmetric Sum = $S_2 = ab+ac+ad+bc+bd+cd$

3rd Symmetric Sum = $S_3 = abc+abd+acd+bcd$

4th Symmetric Sum = $S_4 = abcd$

Notation

The first elementary symmetric sum of $f(x)$ is often written $\sum_{sym}f(x)$. The $n$th can be written $\sum_{sym}^{n}f(x)$

Uses

Any symmetric sum can be written as a polynomial of the elementary symmetric sum functions. For example, $x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = S_1^3 - 3S_1S_2 + 3S_3$. This is often used to solve systems of equations involving sums of powers, combined with Vieta's formulas.

Elementary symmetric sums show up in Vieta's formulas. In a monic polynomial of degree $n$, the coefficient of the $x^n$ term is $(-1)^nS_n$, and the coefficient of the $x^k$ term is $(-1)^{n-k}S_{n-k}$, where the symmetric sums are taken over the roots of the polynomial.

See Also

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