Elementary symmetric sum

An elementary symmetric sum is a type of summation.

Definition

The $k$-th elementary symmetric sum of a set of $n$ numbers is the sum of all products of $k$ of those numbers ($1 \leq k \leq n$). For example, if $n = 4$, and our set of numbers is $\{a, b, c, d\}$, then:

1st Symmetric Sum = $S_1 = a+b+c+d$

2nd Symmetric Sum = $S_2 = ab+ac+ad+bc+bd+cd$

3rd Symmetric Sum = $S_3 = abc+abd+acd+bcd$

4th Symmetric Sum = $S_4 = abcd$

Notation

The first elementary symmetric sum of $f(x)$ is often written $\sum_{sym}f(x)$. The $n$th can be written $\sum_{sym}^{n}f(x)$

Uses

Any symmetric sum can be written as a polynomial of the elementary symmetric sum functions. For example, $x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - xz) + 3xyz = S_1^3 - 3S_1S_2 + 3S_3$. 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, the coefficient of the $x^1$ term is $e_1$, and the coefficient of the $x^k$ term is $e_k$, where the symmetric sums are taken over the roots of the polynomial.

See Also

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