Elementary Symmetric Polynomial - Definition

Definition

The elementary symmetric polynomials in variables X1, …, Xn, written ek(X1, …, Xn) for k = 0, 1, ..., n, can be defined as

\begin{align} e_0 (X_1, X_2, \dots,X_n) &= 1,\\ e_1 (X_1, X_2, \dots,X_n) &= \textstyle\sum_{1 \leq j \leq n} X_j,\\ e_2 (X_1, X_2, \dots,X_n) &= \textstyle\sum_{1 \leq j < k \leq n} X_j X_k,\\ e_3 (X_1, X_2, \dots,X_n) &= \textstyle\sum_{1 \leq j < k < l \leq n} X_j X_k X_l,\\ \end{align}

and so forth, down to

(sometimes the notation σk is used instead of ek). In general, for k ≥ 0 we define

Thus, for each positive integer less than or equal to, there exists exactly one elementary symmetric polynomial of degree in variables. To form the one which has degree, we form all products of -subsets of the variables and add up these terms.

The fact that and so forth is the defining feature of commutative algebra. That is, the polynomial ring formed by taking all linear combinations of products of the elementary symmetric polynomials is a commutative ring.

Read more about this topic:  Elementary Symmetric Polynomial

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