Elementary symmetric sum
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An elementary symmetric sum is a type of summation.
Contents |
Definition
The 
-th elmentary 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 elmentary 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.
