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 [https://en.wikipedia.org/wiki/Sums_of_powers sums of powers], 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 [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, the coefficient of the <math>x^ | + | 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== |
Revision as of 17:32, 25 August 2021
An elementary symmetric sum is a type of summation.
Contents
[hide]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.