elementary symmetric polynomial
ElementarySymmetricPolynomial
2002-01-05 05:13:21
2006-10-22 01:15:16
Definition
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The coefficient of $x^{n-k}$ in the polynomial $(x+t_1) (x+t_2) \cdots (x+t_n)$ is called the $k^\mathrm{th}$ \emph{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:
\begin{description}
\item[$n=1$:]
$
\begin{array}{l}
t_1
\end{array}
$
\item[$n=2$:]
$
\begin{array}{l}
t_1 + t_2\\
t_1 t_2
\end{array}
$
\item[$n=3$:]
$
\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}
$
\end{description}