Algebra/Chapter 10/Symmetric Polynomials/Elementary Symmetric Polynomials

Let there be a polynomial of degree $n \geq 1$

P (z) = \sum_{k = 0}^{n} a_{k} z^{k} \in ℂ [z]

By the Fundamental Theorem of Algebra, it has $n$ complex roots (with multiplicity). Then we can write:

P (z) = a_{n} \prod_{k = 1}^{n} (z - z_{k})

As we know, Vieta's formulae link between the coefficients and roots of a polynomial:

{\begin{matrix} z_{1} + \dots + z_{n} = - \frac{a_{n - 1}}{a_{n}} \\ (z_{1} z_{2} + \dots + z_{1} z_{n}) + (z_{2} z_{3} + \dots + z_{2} z_{n}) + \dots + z_{n - 1} z_{n} = \frac{a_{n - 2}}{a_{n}} \\ (z_{1} z_{2} z_{3} + \dots + z_{1} z_{2} z_{n}) + (z_{1} z_{3} z_{4} + \dots + z_{1} z_{3} z_{n}) + \dots + (z_{2} z_{3} z_{4} + \dots + z_{2} z_{n - 1} z_{n}) + \dots + z_{n - 2} z_{n - 1} z_{n} = - \frac{a_{n - 3}}{a_{n}} \\ ⋮ \\ z_{1} \dots z_{n} = (- 1)^{n} \frac{a_{0}}{a_{n}} \end{matrix}

As we can see, these sums are symmetric polynomial, and are called elementary symmetric polynomials.

Definition

The elementary symmetric polynomials in variables $X_{1}, \dots, X_{n}$ , are defined as such:

\begin{matrix} E_{1} (\vec{X}^{n}) & = \sum_{1 \leq i \leq n} X_{i} \\ E_{2} (\vec{X}^{n}) & = \sum_{1 \leq i_{1} < i_{2} \leq n} X_{i_{1}} X_{i_{2}} \\ ⋮ \\ E_{k} (\vec{X}^{n}) & = \sum_{1 \leq i_{1} < \dots < i_{k} \leq n} X_{i_{1}} \dots X_{i_{k}} \\ ⋮ \\ E_{n} (\vec{X}^{n}) & = X_{1} \dots X_{n} \end{matrix}