Elementary symmetric polynomial

The elementary symmetric polynomials in

x_{1}, \dots, x_{n}

are the sums of all products of

k

distinct variables, for

k = 1, \dots, n

.

Table of contents
- Fundamental theorem of symmetric polynomials Elementary symmetric polynomial
- Newton identities Elementary symmetric polynomial

Fundamental theorem of symmetric polynomials

Every symmetric polynomial over a commutative ring has a unique expression as a polynomial in the elementary symmetric polynomials.

Newton identities

Newton's identities relate the elementary symmetric polynomials

e_{k}

to the power sums

p_{k} = \sum_{i} x_{i}^{k}

. The first three give

e_{1} = p_{1}, 2 e_{2} = p_{1}^{2} - p_{2}, 6 e_{3} = p_{1}^{3} - 3 p_{1} p_{2} + 2 p_{3} .

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Elementary symmetric polynomials are a fundamental class of symmetric polynomials in algebra. Given a set of \( n \) variables, \( x_1, x_2, ..., x_n \), the elementary symmetric polynomials are defined as follows: 1. The first elementary symmetric polynomial \( e_1(x_1, x_2, ...

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