---
tag: SD3M01318E
kind: definition
name: Homogeneous polynomial
msc: [13F20, "12E10", 13A02]
defines: 2C3PAQ7H50
uses: [0CES91M79A, 0DPF6WRZW6]
sources:
  - url: https://en.wikipedia.org/wiki/Homogeneous_polynomial
---

## Definition

Let $R$ be a commutative ring and let $f \in R[x_1, \ldots, x_n]$ be a [[polynomial in several variables|0CES91M79A]].
The total degree of a monomial $x_1^{a_1} \cdots x_n^{a_n}$ is $a_1 + \cdots + a_n$.
Let $d \ge 0$ be an integer.
The polynomial $f$ is *homogeneous of degree $d$* if every monomial that appears in $f$ with a nonzero coefficient has total degree $d$.
The zero polynomial is homogeneous of every degree.

## Remarks

Example: $f = x^2 y + 3 y z^2 - z^3$ in $\Q[x, y, z]$ is homogeneous of degree $3$.
Check: the monomials $x^2 y$, $y z^2$ and $z^3$ have total degrees $2 + 1$, $1 + 2$ and $3$.

Non-example: $g = x^2 + y$ is not homogeneous, because its monomials have total degrees $2$ and $1$.

The polynomial $f$ is homogeneous of degree $d$ if and only if $f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n)$ in $R[\lambda, x_1, \ldots, x_n]$.
Every polynomial is in a unique way a finite sum $f = f_0 + f_1 + \cdots$ where each $f_d$ is homogeneous of degree $d$; these are the homogeneous components of $f$.
A homogeneous polynomial is also called a form; forms of degree $1$, $2$ and $3$ are linear, quadratic and cubic forms.
In one variable, the nonzero homogeneous polynomials of degree $d$ are the $c x^d$ with $c \ne 0$, whose [[degree|0DPF6WRZW6]] is $d$.