Homogeneous polynomials are the algebraic expressions whose terms all have the same total degree. They appear whenever a problem is insensitive to scale: projective geometry identifies proportional coordinates, tangent cones record the first nonzero part of a local equation, and algebraic geometry studies zero sets that are stable under dilation. A general [polynomial](/page/Polynomial) mixes several degrees at once, but a homogeneous polynomial isolates one layer. That single-layer structure is what makes it compatible with projective space, with [vector spaces](/page/Vector%20Space), and with graded algebra.

A useful way to remember the concept is this: a homogeneous polynomial behaves like a power under rescaling. If all variables are multiplied by the same scalar, the value is multiplied by a fixed power of that scalar. This turns a polynomial identity into a geometric scaling law, and it is the reason homogeneous polynomials define well-behaved equations on projective coordinates.

## Definition

The basic setting is a [polynomial ring](/page/Polynomial%20Ring) in finitely many variables over a field. The total degree of a monomial measures how much that monomial scales when all variables are scaled at once, so the definition singles out polynomials whose monomials have a common total degree.

[definition: Homogeneous Polynomial] Let $k$ be a field, let $n \in \mathbb{N}$, and let $k[x_1, \ldots, x_n]$ be the polynomial ring in $n$ variables. A polynomial $f \in k[x_1, \ldots, x_n]$ is homogeneous of degree $d \in \mathbb{N} \cup \{0\}$ if it can be written as \begin{align*} f(x_1, \ldots, x_n) = \sum_{\alpha_1 + \cdots + \alpha_n = d} c_\alpha x_1^{\alpha_1}\cdots x_n^{\alpha_n}, \end{align*} where $c_\alpha \in k$ and each $\alpha_i \in \mathbb{N} \cup \{0\}$. [/definition]

Constant polynomials are included as homogeneous polynomials of degree $0$. The zero polynomial is usually allowed to belong to every homogeneous degree piece when one studies graded rings, although it is not assigned a single ordinary degree.

## Associated Constructions

A homogeneous equation has more structure than an arbitrary polynomial equation, because its solution set is preserved by scalar multiplication. The first geometric object it cuts out lives in affine space, where the zero set is a cone with vertex at the origin.

[definition: Homogeneous Hypersurface] Let $k$ be a field, let $n \in \mathbb{N}$, let $f \in k[x_1, \ldots, x_n]$ be a nonzero homogeneous polynomial, and let $\mathbb{A}^n_k$ denote affine $n$-space over $k$. The affine homogeneous hypersurface defined by $f$ is \begin{align*} V(f) = \{a \in k^n : f(a) = 0\}. \end{align*} [/definition]

Once a homogeneous affine equation cuts out entire scalar lines through the origin, the natural next question is which directions those lines represent. Projective space is built to answer that question: it identifies nonzero scalar multiples, so an equation on projective points must have a zero condition that survives changing representatives. Positive-degree homogeneous polynomials provide exactly that invariant zero condition.

[definition: Projective Homogeneous Hypersurface] Let $k$ be a field, let $n \in \mathbb{N}$ with $n \ge 1$, and let $f \in k[x_1, \ldots, x_n]$ be a nonzero homogeneous polynomial of degree $d > 0$. The projective homogeneous hypersurface defined by $f$ is \begin{align*} Z(f) = \{[a_1 : \cdots : a_n] \in \mathbb{P}^{n-1}_k : f(a_1, \ldots, a_n) = 0\}. \end{align*} [/definition]

This definition records the $k$-rational projective points of the hypersurface. In algebraic geometry, the same homogeneous equation is often studied after extending scalars to an [algebraic closure](/page/Algebraic%20Closure) of $k$, or as a projective scheme or functor of points; the representative-independence argument is the same in each setting.

The projective definition is useful because it turns an affine cone into an equation on directions: replacing $a$ by $\lambda a$ rescales $f(a)$ by a power of $\lambda$, so vanishing is independent of the chosen representative. To apply this idea to an arbitrary affine polynomial, one first has to separate the polynomial into its degree layers and identify which pieces already have this scaling behaviour.

A mixed polynomial can contain terms with several different scaling behaviours at once. To compare those behaviours, one needs a precise way to isolate only the terms of a chosen total degree while leaving the rest of the polynomial available for other arguments.

[definition: Homogeneous Component] Let $k$ be a field, let $n \in \mathbb{N}$, and let $d \in \mathbb{N} \cup \{0\}$. Write $k[x_1, \ldots, x_n]_d$ for the $k$-vector space of homogeneous polynomials of degree $d$ in $k[x_1, \ldots, x_n]$. The degree-$d$ homogeneous component extraction map is the $k$-linear projection \begin{align*} \pi_d : k[x_1, \ldots, x_n] \to k[x_1, \ldots, x_n]_d \end{align*} that sends a polynomial $f$ to the sum of all monomial terms of $f$ whose total degree is $d$. This image is denoted $f_d$. [/definition]

The homogeneous component lets a nonhomogeneous polynomial be separated into layers. This is useful in local geometry, where the lowest-degree nonzero layer gives the leading approximation near the origin, and in projective constructions, where extra variables are introduced to force all terms to have the same degree.

## Equivalent Characterisations

The monomial definition is concrete, but the scaling characterisation is often the more geometric one. It says that homogeneous polynomials are eigenvectors for the operation of dilating the variables.

[quotetheorem:9252]

The scaling test involves substituting $\lambda x$ into the whole polynomial, which can be inconvenient when one wants a local differential criterion. For a single homogeneous layer, the radial derivative should measure exactly how fast the polynomial changes under dilation, turning homogeneity into an identity involving the usual partial derivatives.