Difference between revisions of "Elementary symmetric sum"
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| + | An '''elementary symmetric sum''' is a type of [[summation]]. | ||
| + | |||
== Definition == | == Definition == | ||
| + | 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: | ||
| − | + | 1st Symmetric Sum = <math>S_1 = a+b+c+d</math> | |
| − | + | 2nd Symmetric Sum = <math>S_2 = ab+ac+ad+bc+bd+cd</math> | |
| − | + | 3rd Symmetric Sum = <math>S_3 = abc+abd+acd+bcd</math> | |
| − | + | 4th Symmetric Sum = <math>S_4 = abcd</math> | |
| − | + | ==Notation== | |
| + | 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 == | ||
| + | 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. | ||
| + | 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== |
| + | *[[Symmetric sum]] | ||
| + | *[[Cyclic sum]] | ||
| − | + | [[Category:Algebra]] | |
| + | [[Category:Definition]] | ||
Latest revision as of 11:51, 26 July 2023
An elementary symmetric sum is a type of summation.
Contents
Definition
The 
-th elementary symmetric sum of a set of 
numbers is the sum of all products of of those numbers (
). For example, if 
, and our set of numbers is 
, then:
1st Symmetric Sum = 
2nd Symmetric Sum = 
3rd Symmetric Sum = 
4th Symmetric Sum = 
Notation
The first elementary symmetric sum of 
is often written 
. The th can be written 
Uses
Any symmetric sum can be written as a polynomial of the elementary symmetric sum functions. For example, 
. 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 , the coefficient of the 
term is 
, and the coefficient of the 
term is 
, where the symmetric sums are taken over the roots of the polynomial.
