# Difference between revisions of "Elementary symmetric sum"

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==Notation== | ==Notation== | ||

− | The first | + | 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 == |

## Revision as of 00:44, 28 November 2014

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 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.