# Difference between revisions of "Elementary symmetric sum"

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== Definition == | == Definition == | ||

− | The <math>k</math>-th ''' | + | 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> | 1st Symmetric Sum = <math>S_1 = a+b+c+d</math> |

## Revision as of 00:45, 28 November 2014

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.