Difference between revisions of "Elementary symmetric sum"
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− | 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 [[power sum]]s, combined with Vieta's formulas. | + | 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 [[power sum]]s, combined with Vieta's formulas. |
Elementary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial, the coefficient of the <math>x^1</math> term is <math>e_1</math>, and the coefficient of the <math>x^k</math> term is <math>e_k</math>, where the symmetric sums are taken over the roots of the polynomial. | Elementary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial, the coefficient of the <math>x^1</math> term is <math>e_1</math>, and the coefficient of the <math>x^k</math> term is <math>e_k</math>, where the symmetric sums are taken over the roots of the polynomial. |
Revision as of 17:22, 21 January 2016
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.