---
tag: 0CES91M79A
kind: definition
name: Polynomials in several variables
msc: [13F20]
defines: 07GNX8GCYP
uses: [0Q9EZS90GY, 0GYGHKX834]
sources:
  - url: https://en.wikipedia.org/wiki/Polynomial_ring
    locator: Definition (multivariate case)
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$ be an integer.
For $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n$, the monomial $x^\alpha$ is the formal product $x_1^{\alpha_1} \cdots x_n^{\alpha_n}$.
A polynomial in $x_1, \dots, x_n$ over $R$ is a finite sum $\sum_\alpha c_\alpha x^\alpha$ with $c_\alpha \in R$.
Two polynomials are equal if and only if they have the same coefficient $c_\alpha$ for each $\alpha$.
The ring $R[x_1, \dots, x_n]$ is the set of these polynomials, with coefficientwise addition and the product
$$\Big(\sum_\alpha c_\alpha x^\alpha\Big) \Big(\sum_\beta d_\beta x^\beta\Big) = \sum_\gamma \Big(\sum_{\alpha + \beta = \gamma} c_\alpha d_\beta\Big) x^\gamma.$$

## Remarks

There is a natural isomorphism $R[x_1, \dots, x_n] \cong R[x_1, \dots, x_{n-1}][x_n]$. So the ring can also be built by repeated use of the [[polynomial ring|0Q9EZS90GY]] in one variable.
The same construction works for any set of variables, with exponent maps that are zero at all but finitely many variables.
The polynomial $x_1 x_2 + x_3^2$ is an element of $\mathbb{Z}[x_1, x_2, x_3]$.

## Explanations

### L1 human

A polynomial in several variables is a finite sum of terms like $5 x^2 y z^3$.
You add and multiply such polynomials with the usual rules of algebra.
Each term is a coefficient times a product of powers of the variables.