elementary symmetric polynomial

The coefficient of $x^{n-k}$ in the polynomial $(x+t_{1})(x+t_{2})\cdots(x+t_{n})$ is called the $k^{\mathrm{th}}$ elementary symmetric polynomial in the $n$ variables $t_{1},\dots,t_{n}$. The elementary symmetric polynomials can also be constructed by taking the sum of all possible degree $k$ monomials in $t_{1},\dots,t_{n}$ having distinct factors.

The first few examples are:

$n=1$:

$\begin{array}[]{l}t_{1}\end{array}$

$n=2$:

$\begin{array}[]{l}t_{1}+t_{2}\\ t_{1}t_{2}\end{array}$

$n=3$:

$\begin{array}[]{l}t_{1}+t_{2}+t_{3}\\ t_{1}t_{2}+t_{2}t_{3}+t_{1}t_{3}\\ t_{1}t_{2}t_{3}\end{array}$

Title elementary symmetric polynomial ElementarySymmetricPolynomial 2013-03-22 12:09:01 2013-03-22 12:09:01 djao (24) djao (24) 9 djao (24) Definition msc 05E05