Difference between revisions of "Elementary symmetric sum"
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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 [[power sum]]s, combined with Vieta's formulas. | + | 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 [[power sum]]s, combined with Vieta's formulas. |
Elementary 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>, where the symmetric sums are taken over the roots of the polynomial. | Elementary 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>, where the symmetric sums are taken over the roots of the polynomial. | ||
Revision as of 16:22, 21 January 2016
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 power sums, combined with Vieta's formulas.
Elementary symmetric sums show up in Vieta's formulas. In a monic polynomial, 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.
