PlanetMath: elementary symmetric polynomial

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elementary symmetric polynomial

(Definition)

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 variables $t_1, \dots, t_n$ . The elementary symmetric polynomials can also be constructed by taking the sum of all possible degree monomials in $t_1,\dots, t_n$ having distinct factors.

The first few examples are:

:: $\begin{displaymath} \begin{array}{l} t_1 \end{array}\end{displaymath}$
:: $\begin{displaymath} \begin{array}{l} t_1 + t_2\ t_1 t_2 \end{array}\end{displaymath}$
:: $\begin{displaymath} \begin{array}{l} t_1 + t_2 + t_3\ t_1 t_2 + t_2 t_3 + t_1 t_3\ t_1 t_2 t_3 \end{array}\end{displaymath}$

"elementary symmetric polynomial" is owned by djao. [ full author list (2) ]

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Attachments:: algebraic independence of elementary symmetric polynomials (Theorem) by mclase
elementary symmetric polynomial in terms of power sums (Result) by rspuzio

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Cross-references: factors, monomials, degree, sum, variables, polynomial, coefficient
There are 13 references to this entry.

This is version 5 of elementary symmetric polynomial, born on 2002-01-05, modified 2006-10-22.
Object id is 1340, canonical name is ElementarySymmetricPolynomial.
Accessed 5410 times total.

Classification:

AMS MSC:

05E05 (Combinatorics :: Algebraic combinatorics :: Symmetric functions)

Pending Errata and Addenda

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