elementary symmetric polynomial

The coefficient of $x^{n-k}$ in the polynomial $(x+t_1)(x+t_2)\mathrm{\cdots }(x+t_n)$ is called the $k^{\mathrm{th}}$ elementary symmetric polynomial in the $n$ variables $t_1,\mathrm{\dots },t_n$. The elementary symmetric polynomials can also be constructed by taking the sum of all possible degree $k$ monomials in $t_1,\mathrm{\dots },t_n$ having distinct factors.

The first few examples are:

$n=1$:

$\begin{array}{c}{t}_1\hfill \end{array}$

$n=2$:

$\begin{array}{c}{t}_1+t_2\hfill \\ t_1{t}_2\hfill \end{array}$

$n=3$:

$\begin{array}{c}{t}_1+t_2+t_3\hfill \\ t_1{t}_2+t_2{t}_3+t_1{t}_3\hfill \\ t_1{t}_2{t}_3\hfill \end{array}$

Title	elementary symmetric polynomial
Canonical name	ElementarySymmetricPolynomial
Date of creation	2013-03-22 12:09:01
Last modified on	2013-03-22 12:09:01
Owner	djao (24)
Last modified by	djao (24)
Numerical id	9
Author	djao (24)
Entry type	Definition
Classification	msc 05E05