Elementary symmetric polynomial

For $n\geq 1$, and $1\leq k\leq n$, the $k$-th elementary symmetric polynomial is the 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 :
  $$\sigma_{1}=X+Y+Z,\quad\sigma_{2}=XY+YZ+XZ,\quad \sigma_{3}=XYZ.$$