Symmetric polynomial

Definition

Let $R$ be a commutative ring and let $n \ge 1$. A polynomial $f \in R[x_1, \dots, x_n]$ is symmetric if $$f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = f(x_1, \dots, x_n)$$ for every permutation $\sigma$ of $\{1, \dots, n\}$.

Remarks

Examples in two variables: $x_1 + x_2$, $x_1 x_2$ and $x_1^2 + x_2^2$. The polynomial $x_1 - x_2$ is not symmetric. The symmetric polynomials form a subring of $R[x_1, \dots, x_n]$, the ring of invariants of the symmetric group. The main examples are the elementary symmetric polynomials and the power sums. The coefficients of a monic polynomial are symmetric polynomials in its roots, by Vieta's formulas.

Explanations

L1 human

A symmetric polynomial treats all its variables in the same way. Swapping two variables gives the same polynomial. This is why symmetric expressions in the roots of a polynomial can be computed from the coefficients.