# 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> | + | 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== | ||

− | 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 == | ||

− | Any symmetric sum can be written as a [[polynomial]] of the | + | Any [[symmetric sum]] can be written as a [[polynomial]] of the elementary symmetric sum functions. For example, <math>x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = S_1^3 - 3S_1S_2 + 3S_3</math>. This is often used to solve systems of equations involving [https://en.wikipedia.org/wiki/Sums_of_powers sums of powers], combined with Vieta's formulas. |

− | + | Elementary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial of degree <math>n</math>, the coefficient of the <math>x^0</math> term is <math>(-1)^nS_n</math>, and the coefficient of the <math>x^k</math> term is <math>(-1)^{n-k}S_{n-k}</math>, where the symmetric sums are taken over the roots of the polynomial. | |

==See Also== | ==See Also== | ||

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*[[Cyclic sum]] | *[[Cyclic sum]] | ||

− | [[Category: | + | [[Category:Algebra]] |

[[Category:Definition]] | [[Category:Definition]] |

## Latest revision as of 17:32, 25 August 2021

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 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.