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elementary symmetric polynomial
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(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 $k^\mathrm{th}$ 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} $
- $n=2$
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$ \begin{array}{l} t_1 + t_2\\ t_1 t_2 \end{array} $
- $n=3$
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$ \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} $
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"elementary symmetric polynomial" is owned by djao. [ full author list (2) ]
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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 6920 times total.
Classification:
| AMS MSC: | 05E05 (Combinatorics :: Algebraic combinatorics :: Symmetric functions) |
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Pending Errata and Addenda
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