# Difference between revisions of "Elementary symmetric sum"

m (→Uses: spelling errors, plus what we sum over in Vieta's formulas.) |
m (The more popular notation is S_n for sums) |
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The <math>k</math>-th '''elmentary 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: | The <math>k</math>-th '''elmentary 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> | + | 1st Symmetric Sum = <math>S_1 = a+b+c+d</math> |

− | 2nd Symmetric Sum = <math> | + | 2nd Symmetric Sum = <math>S_2 = ab+ac+ad+bc+bd+cd</math> |

− | 3rd Symmetric Sum = <math> | + | 3rd Symmetric Sum = <math>S_3 = abc+abd+acd+bcd</math> |

− | 4th Symmetric Sum = <math> | + | 4th Symmetric Sum = <math>S_4 = abcd</math> |

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

## Revision as of 23:02, 27 June 2013

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.