Definition:Elementary Symmetric Function

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Definition

Let $a, b \in \Z$ be integers such that $b \ge a$.

Let $U$ be a set of $n = b - a + 1$ numbers $\left\{ {x_a, x_{a + 1}, \ldots, x_b}\right\}$.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.


An elementary symmetric function of degree $m$ is a polynomial which can be defined by the formula:

\(\displaystyle e_m \left({U}\right)\) \(=\) \(\displaystyle \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} \left({\prod_{i \mathop = 1}^m x_{j_i} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} x_{j_1} x_{j_2} \cdots x_{j_m}\)

That is, it is the sum of all products of $m$ distinct elements of $\left\{ {x_a, x_{a + 1}, \ldots, x_b}\right\}$.


Examples

Example: $m = 0$

$e_0 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right) = 1$


Example: $m = 1$

\(\displaystyle e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\displaystyle x_1 + x_2 + \cdots + x_n\)


Example: $m = 2$

\(\displaystyle e_2 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\displaystyle x_1 x_2 + x_1 x_3 + \cdots + x_1 x_n\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle x_2 x_3 + \cdots + x_2 x_n\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \cdots\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle x_{n - 1} x_n\)


Example: $m = n$

\(\displaystyle e_n \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\displaystyle x_1 x_2 \cdots x_n\)


Example: $m > n$

Let $m > n$.

Then:

\(\displaystyle e_m \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\displaystyle 0\)


Example: Monic polynomial coefficients

Let $\set {x_1, x_2, \ldots, x_n}$ be a set of real or complex values, not required to be unique.

The expansion of the monic polynomial in variable $x$ with roots $\set {x_1, x_2, \ldots, x_n}$ has coefficients which are sign factors times an elementary symmetric function:

$\displaystyle \prod_{j \mathop = 1}^n \paren {x - x_j} = x^n - e_1 \paren {\set {x_1, \ldots, x_n} } x^{n - 1} + e_2 \paren {\set {x_1, \ldots, x_n} } x^{n - 2} + \cdots + \paren {-1}^n e_n \paren {\set {x_1, \ldots, x_n} }$


Example: Recursion

Let $\set {z_1, z_2, \ldots, z_{n+1}}$ be a set of $n+1$ values, duplicate values permitted.

Then for $1 \le m \le n$:

$\displaystyle e_m \paren { \set {z_1,\ldots,z_n,z_{n+1} } } = z_{n+1} e_{m-1} \paren { \set {z_1,\ldots,z_n} } + e_m \paren { \set {z_1,\ldots,z_n} } $


Also known as

An elementary symmetric function is also known as an elementary symmetric polynomial.


Also see

  • Results about elementary symmetric functions can be found here.


Sources