Elementary Symmetric Function/Examples

Examples of Elementary Symmetric Function

Example: $m = 0$

$\map {e_0} {\set {x_1, x_2, \ldots, x_n} } = 1$

Example: $m = 1$

\(\ds e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\)

\(=\)

\(\ds x_1 + x_2 + \cdots + x_n\)

Example: $m = 2$

\(\ds e_2 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\)

\(=\)

\(\ds x_1 x_2 + x_1 x_3 + \cdots + x_1 x_n\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds x_2 x_3 + \cdots + x_2 x_n\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \cdots\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds x_{n - 1} x_n\)

Example: $m = n$

\(\ds \map {e_n} {\set {x_1, x_2, \ldots, x_n} }\)

\(=\)

\(\ds x_1 x_2 \cdots x_n\)

Example: $m > n$

Let $m > n$.

Then:

\(\ds \map {e_m} {\set {x_1, x_2, \ldots, x_n} }\)

\(=\)

\(\ds 0\)

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:

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