Elementary symmetric polynomial

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7/8/2026, 2:16:27 PM · Ancient Tree

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For $n\geq 1$, and $1\leq k\leq n$, the $k$-th elementary symmetric polynomial is the [[Symmetric polynomial|symmetric polynomial]] defined as :

2$$\sigma_{n,k}(X_{1},\ldots,X_{n})=\sum_{1 \leqslant j_1<\cdots<j_k \leqslant n} X_{j_1} \ldots X_{j _{k}}$$

3##### Remarks and examples

4- The $n$ index is often ommitted when the context is clear, i.e., $\sigma_{n,k}=\sigma_{k}$

5- By convention, $\sigma_{0}=1$.

6- For $n=3$, the elementary symmetric polynomials are given by :

7$$

8\sigma_{1}=X+Y+Z,\quad\sigma_{2}=XY+YZ+XZ,\quad \sigma_{3}=XYZ.$$

Revision 349

7/8/2026, 2:15:01 PM · Ancient Tree

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For $n\geq 1$, and $1\leq k\leq n$, the $k$-th elementary symmetric polynomial is the [[Symmetric polynomial|symmetric polynomial]] defined as :
$$\sigma_{n,k}(X_{1},\ldots,X_{n})=\sum_{1 \leqslant j_1<\cdots<j_k \leqslant n} X_{j_1} \ldots X_{j _{k}}$$
##### Remarks and examples
- The $n$ index is often ommitted when the context is clear, i.e., $\sigma_{n,k}=\sigma_{k}$
- By convention, $\sigma_{0}=1$.
- For $n=3$, the elementary symmetric polynomials are given by :
$$
\sigma_{1}=X+Y+Z,\quad\sigma_{2}=XY+YZ+XZ,\quad \sigma_{3}=XYZ.$$