Elementary symmetric sum
An elementary symmetric sum is a type of summation.
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 =
The first elementary symmetric sum of is often written . The th can be written
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.