Homogeneous Polynomial¶

Definition¶

A polynomial on $\mathbb{R}^n$ is called homogeneous of degree $k$ if all of its nonzero terms have degree $k$ . Such a polynomial is also called a (algebraic) form.

Every degree- $k$ monomial acquires the same factor $c^k$ when its variables are scaled by $c$ . Their sum therefore satisfies the defining identity of a Homogeneous Function.

Interpretation¶

A homogeneous polynomial is a polynomial in which every nonzero term has the same total degree $k$ . Scaling all variables by $c$ then multiplies every term, and hence the whole polynomial, by $c^k$ :

P(cx_1,\dots,cx_n)=c^kP(x_1,\dots,x_n).

So a homogeneous polynomial of degree $k$ is a homogeneous function of degree $k$ . Conversely, a polynomial that satisfies this identity for every $c\neq0$ is homogeneous of degree $k$ , so for polynomials the two notions agree.

To check homogeneity, add the exponents in each nonzero term: $x^2y+y^3$ is homogeneous of degree $3$ , while $x^2+y$ has terms of degrees $2$ and $1$ and is not homogeneous. Every polynomial on $\mathbb{R}^n$ splits uniquely into homogeneous parts by grouping its terms by total degree; for example, $x^2+2xy+y^2+x+1$ has the parts $x^2+2xy+y^2$ $y^{2}$ , $x$ and $1$ .

The forms of low degree are familiar from linear algebra. Forms of degree $1$ are the linear functions $a_1x_1+\cdots+a_nx_n$ $a_{n} x_{n}$ , that is, the linear maps $\mathbb{R}^n\to\mathbb{R}$ $R$ . Forms of degree $2$ are the quadratic forms $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}$ .

A common slip is to compare the degree of each variable instead of the total degree. The terms $x^3$ and $xy^2$ look different, but both have total degree $3$ , so $x^3-3xy^2$ is homogeneous of degree $3$ .

Key Results¶

The sum of two homogeneous polynomials of the same degree is homogeneous of that degree.
The product of a homogeneous polynomial of degree $k$ and one of degree $\ell$ is homogeneous of degree $k+\ell$ $ℓ$ .

Examples¶

$P(x) = 2x^5$ is a homogeneous polynomial of degree 5.
$P(x,y) = x^2 + xy + y^2$ is homogeneous of degree 2. Each term has total degree 2.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 7

Aliases: algebraic form, quantic Keywords: form Symbols: real numbers

Last updated: 20 October 2026 03:02