Definition:Symmetric Function/Elementary
Definition
Let $a, b \in \Z$ be integers such that $b \ge a$.
Let $U$ be a set of $n = b - a + 1$ numbers $\set {x_a, x_{a + 1}, \ldots, x_b}$.
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:
| \(\ds \map {e_m} U\) | \(=\) | \(\ds \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} \paren {\prod_{i \mathop = 1}^m x_{j_i} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} x_{j_1} x_{j_2} \cdots x_{j_m}\) |
That is, it is the sum of all products of $m$ distinct elements of $\set {x_a, x_{a + 1}, \dotsc, x_b}$.
Examples
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:
- $\ds \prod_{j \mathop = 1}^n \paren {x - x_j} = x^n - \map {e_1} {\set {x_1, \ldots, x_n} } x^{n - 1} + \map {e_2} {\set {x_1, \ldots, x_n} } x^{n - 2} + \dotsb + \paren {-1}^n \map {e_n} {\set {x_1, \ldots, x_n} }$
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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): symmetric function: 2.
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: Exercise $10$