---
tag: 06CTFY2DEZ
kind: definition
name: Symmetric polynomial
msc: ["05E05"]
defines: 0VHSHB9B6J
uses: [0CES91M79A]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_polynomial
    locator: lead section
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] 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|0ZMY8441GW]] and the [[power sums|0AS1FZT6BM]].
The coefficients of a monic polynomial are symmetric polynomials in its roots, by [[Vieta's formulas|0M8EWQJ1YV]].

## 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.