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. This leads to a practical question: can homogeneity be detected without comparing all dilations at once? [Euler's criterion](/theorems/1715) answers this by replacing the global scaling law with a first-order identity involving the variables and partial derivatives. [quotetheorem:9253] The characteristic assumption avoids the possibility that a positive degree becomes zero inside the field. In characteristic $p$, the polynomial $x^p$ has derivative $0$, so Euler's identity must be stated with care. For parameter counts and linear systems, it is not enough to know that a polynomial is homogeneous. We need the collection of all homogeneous polynomials of a fixed degree as a vector space in its own right. [definition: Space of Homogeneous Polynomials] Let $k$ be a field, let $n \in \mathbb{N}$, and let $d \in \mathbb{N} \cup \{0\}$. The space of homogeneous polynomials of degree $d$ in $n$ variables is \begin{align*} k[x_1, \ldots, x_n]_d = \{f \in k[x_1, \ldots, x_n] : f \text{ is homogeneous of degree } d\}. \end{align*} [/definition] This notation records the grading of the polynomial ring: \begin{align*} k[x_1, \ldots, x_n] = \bigoplus_{d=0}^{\infty} k[x_1, \ldots, x_n]_d. \end{align*} Each summand is finite-dimensional, while the full polynomial ring is infinite-dimensional. ## Examples The simplest homogeneous polynomials are familiar quadratic and cubic forms. They already show the main geometric behaviour: their zero sets are cones in affine space and become projective curves or hypersurfaces after quotienting by scalar multiplication. [example: Quadratic Form in Three Variables] Over $\mathbb{R}$, consider \begin{align*} f(x_1,x_2,x_3)=x_1^2+x_2^2-x_3^2. \end{align*} Each monomial has total degree $2$: $x_1^2$ has exponent sum $2$, $x_2^2$ has exponent sum $2$, and $x_3^2$ has exponent sum $2$. Hence $f$ is homogeneous of degree $2$. For every $\lambda \in \mathbb{R}$, substituting $\lambda x_i$ for each variable gives \begin{align*} f(\lambda x_1,\lambda x_2,\lambda x_3)=(\lambda x_1)^2+(\lambda x_2)^2-(\lambda x_3)^2. \end{align*} Using $(ab)^2=a^2b^2$ in $\mathbb{R}$, this becomes \begin{align*} f(\lambda x_1,\lambda x_2,\lambda x_3)=\lambda^2x_1^2+\lambda^2x_2^2-\lambda^2x_3^2. \end{align*} Factoring out the common scalar $\lambda^2$ gives \begin{align*} f(\lambda x_1,\lambda x_2,\lambda x_3)=\lambda^2(x_1^2+x_2^2-x_3^2)=\lambda^2 f(x_1,x_2,x_3). \end{align*} Thus the affine zero set is \begin{align*} V(f)=\{(x_1,x_2,x_3)\in \mathbb{R}^3 : x_1^2+x_2^2-x_3^2=0\}. \end{align*} Equivalently, it is the cone $x_1^2+x_2^2=x_3^2$. Since scaling a point by $\lambda$ multiplies the value of $f$ by $\lambda^2$, the equation $f=0$ is unchanged after replacing a nonzero representative by a scalar multiple. Therefore the projectivisation is the conic \begin{align*} Z(f)=\{[x_1:x_2:x_3]\in \mathbb{P}^2_\mathbb{R}:x_1^2+x_2^2-x_3^2=0\}. \end{align*} This example shows how a quadratic homogeneous equation cuts out an affine cone and then descends to a well-defined equation on projective directions. [/example] Passing from an affine cone to projective space removes the zero vector, so one has to check whether that point is genuinely part of every positive-degree homogeneous zero set. The scaling law forces vanishing at the origin, making the origin the common vertex that is discarded when only directions are retained. The general issue is not special to the quadratic cone above: any positive-degree homogeneous equation should define a cone with a distinguished vertex at the origin. The next formal statement records this basic geometric consequence of homogeneity. [quotetheorem:9254] Projective space removes this common point and remembers directions instead. The vanishing at the origin is not an accidental feature of a few examples; it is forced by positive-degree homogeneity, so the origin cannot distinguish one nonzero homogeneous form from another. Passing to projective space is therefore the natural way to study the zero sets of forms: one records the lines through the origin on which the form vanishes, while discarding the common zero that every positive-degree form has automatically. A boundary example is just as important. A polynomial can have a well-defined total degree without being homogeneous; what fails is the uniform scaling law. [example: Nonhomogeneous Polynomial with Mixed Degrees] The polynomial \begin{align*} g(x,y) = x^2 + xy + y \end{align*} in $\mathbb{R}[x,y]$ has three monomial terms. The term $x^2$ has total degree $2$, the term $xy=x^1y^1$ has total degree $1+1=2$, and the term $y$ has total degree $1$. Therefore the degree-$2$ homogeneous component is \begin{align*} g_2(x,y)=x^2+xy, \end{align*} and the degree-$1$ homogeneous component is \begin{align*} g_1(x,y)=y. \end{align*} Now substitute $\lambda x$ for $x$ and $\lambda y$ for $y$: \begin{align*} g(\lambda x,\lambda y)=(\lambda x)^2+(\lambda x)(\lambda y)+\lambda y. \end{align*} Using commutativity and associativity in $\mathbb{R}[x,y,\lambda]$, the three terms are \begin{align*} (\lambda x)^2=\lambda^2x^2, \end{align*} \begin{align*} (\lambda x)(\lambda y)=\lambda^2xy, \end{align*} and \begin{align*} \lambda y=\lambda y. \end{align*} Hence \begin{align*} g(\lambda x,\lambda y)=\lambda^2x^2+\lambda^2xy+\lambda y. \end{align*} If $g$ were homogeneous of some degree $d$, then scaling would give \begin{align*} g(\lambda x,\lambda y)=\lambda^d g(x,y)=\lambda^d x^2+\lambda^d xy+\lambda^d y. \end{align*} Comparing the coefficient of $x^2$ forces $d=2$, while comparing the coefficient of $y$ forces $d=1$. These two requirements are incompatible, so no single degree $d$ works. Thus $g$ is not homogeneous: it is a sum of a degree-$2$ part and a degree-$1$ part rather than one pure homogeneous layer. [/example] The failure above is exactly what prevents $g=0$ from defining a projective equation. The point $[2:-1]$ has representatives $(1,-1/2)$ and $(2,-1)$. Evaluating at the first representative gives \begin{align*} g(1,-1/2) = 1 - 1/2 - 1/2 = 0. \end{align*} Evaluating at the second gives \begin{align*} g(2,-1) = 4 - 2 - 1 = 1. \end{align*} Membership in the equation $g=0$ therefore depends on the chosen representative, so the equation is not projectively well-defined. Affine equations often have mixed degree, but projective geometry needs homogeneous equations. Homogenisation is the standard construction that adds one coordinate and raises every term to the same total degree. [definition: Homogenisation] Let $k$ be a field, let $n \in \mathbb{N}$, let $d \in \mathbb{N} \cup \{0\}$, and let $k[x_1, \ldots, x_n]_{\le d}$ denote the $k$-vector space of polynomials of total degree at most $d$. The degree-$d$ homogenisation map sends $f$ to $f^h$: \begin{align*} h_d: k[x_1, \ldots, x_n]_{\le d} \to k[x_0,x_1,\ldots,x_n]_d. \end{align*} It is given by $h_d(f)=f^h$. For \begin{align*} f=\sum_{|\alpha|\le d} c_\alpha x_1^{\alpha_1}\cdots x_n^{\alpha_n}, \end{align*} the image is \begin{align*} f^h=\sum_{|\alpha|\le d} c_\alpha x_0^{d-|\alpha|}x_1^{\alpha_1}\cdots x_n^{\alpha_n}. \end{align*} [/definition] The same construction is often remembered by the expression \begin{align*} f^h(x_0,x_1,\ldots,x_n) = x_0^d f\left(\frac{x_1}{x_0}, \ldots, \frac{x_n}{x_0}\right), \end{align*} with the result expanded as a polynomial. Homogenisation is the standard passage from affine algebraic geometry to projective algebraic geometry. [example: Homogenising a Plane Curve] For \begin{align*} f(x,y)=y-x^2 \end{align*} in $\mathbb{R}[x,y]$, the monomial $y$ has total degree $1$ and the monomial $x^2$ has total degree $2$, so the total degree of $f$ is $2$. To homogenise to degree $2$ with the new variable $z$, each term is multiplied by the power of $z$ needed to make total degree $2$. Thus the degree-$1$ term $y$ becomes $zy$, while the degree-$2$ term $x^2$ remains $x^2$, giving \begin{align*} f^h(z,x,y)=zy-x^2. \end{align*} Equivalently, on the chart where $z\neq 0$, \begin{align*} z^2 f\left(\frac{x}{z},\frac{y}{z}\right)=z^2\left(\frac{y}{z}-\frac{x^2}{z^2}\right). \end{align*} Distributing $z^2$ over the two terms gives \begin{align*} z^2\left(\frac{y}{z}-\frac{x^2}{z^2}\right)=z^2\frac{y}{z}-z^2\frac{x^2}{z^2}. \end{align*} Since $z\neq 0$, cancellation gives \begin{align*} z^2\frac{y}{z}=zy. \end{align*} Similarly, \begin{align*} z^2\frac{x^2}{z^2}=x^2. \end{align*} Therefore \begin{align*} z^2 f\left(\frac{x}{z},\frac{y}{z}\right)=zy-x^2. \end{align*} The affine parabola $y=x^2$ is recovered on the affine chart $z\neq 0$. Indeed, if $zy-x^2=0$ and $z\neq 0$, then after scaling the projective representative by $z^{-1}$ we may take $z=1$, and the equation becomes \begin{align*} 1\cdot y-x^2=0. \end{align*} Thus on this chart the projective equation is exactly \begin{align*} y-x^2=0. \end{align*} The projective closure is the conic \begin{align*} Z(zy-x^2)\subset \mathbb{P}^2_{\mathbb{R}} \end{align*} in coordinates $[z:x:y]$. Points at infinity have $z=0$, so the equation becomes \begin{align*} 0\cdot y-x^2=0. \end{align*} This is \begin{align*} -x^2=0, \end{align*} and over $\mathbb{R}$ this forces $x=0$. A projective point cannot have all coordinates zero, so $y\neq 0$. Hence \begin{align*} [0:0:y]=[0:0:1], \end{align*} after scaling by $y^{-1}$. Thus homogenisation turns the affine parabola into a projective conic by adding exactly one point at infinity, namely $[0:0:1]$. [/example] This example shows that homogenisation is not cosmetic. It records how an affine curve closes up at infinity, which is one of the central reasons projective geometry simplifies the study of intersections. ## Properties The first structural fact is that homogeneous pieces give a direct-sum decomposition. This is the algebraic source of many degree arguments: equations can be compared degree by degree. [quotetheorem:9255] The uniqueness comes from the fact that distinct monomials form a basis of the polynomial ring as a vector space. This makes homogeneous decomposition a reliable tool rather than a choice of notation. Degree decompositions are useful only if algebraic operations do not destroy them. When two pure-degree pieces are multiplied, every resulting monomial should have the summed degree; the key point is to make this compatibility precise and to know when the product still has the expected ordinary degree. [quotetheorem:9256] The field hypothesis ensures that multiplying two nonzero leading forms cannot collapse to the zero polynomial. More generally, over a commutative ring with zero divisors, the product still lies in the degree-$(d+e)$ homogeneous piece, but it may be the zero polynomial and therefore may not have ordinary degree $d+e$. To use homogeneous polynomials as parameters, we need to know how many independent coefficients they have. The following dimension count turns degree-$d$ forms into a concrete finite-dimensional family. [quotetheorem:9257] The [binomial coefficient](/page/Binomial%20Coefficient) counts monomials $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ with $\alpha_1+\cdots+\alpha_n=d$. It is the same stars-and-bars count that appears in symmetric powers. A homogeneous equation scales its values by a predictable power, so a zero should remain a zero after dilation. This is the geometric obstruction that separates homogeneous hypersurfaces from general affine hypersurfaces: their zero sets must be unions of scalar rays before projectivisation. [quotetheorem:9258] This theorem is often the first geometric test for homogeneity. If an affine zero set is not closed under scalar multiplication, it cannot be the zero set of homogeneous equations alone. ## Relationship to Other Concepts Homogeneous polynomials are the coordinate-level language of projective geometry. Since a point of projective space is an equivalence class $[a_1:\cdots:a_n]$ of nonzero vectors up to scalar multiple, a homogeneous equation $f(a_1,\ldots,a_n)=0$ is well-defined on projective points. Nonhomogeneous equations do not provide this invariance in general, even if special examples can have accidentally scale-invariant zero sets. This is why projective varieties are defined by homogeneous ideals rather than arbitrary ideals. They are also the polynomial analogue of homogeneous functions. In analysis, a function $F: \mathbb{R}^n \to \mathbb{R}$ is homogeneous of degree $d$ if $F(\lambda x)=\lambda^d F(x)$ for positive scalars $\lambda$. Homogeneous polynomials are the algebraic examples for which the scaling law is encoded by finitely many monomials. In local geometry, the first nonzero homogeneous component of a polynomial at a point describes the tangent cone. If an affine polynomial $f$ vanishes at the origin and begins with a degree-$m$ homogeneous part $f_m$, then $f_m=0$ gives the leading-order cone of directions in which the hypersurface approaches the origin. This connects homogeneous polynomials to tangent spaces, singularities, and algebraic multiplicity. Homogeneous polynomials also form symmetric tensors in disguise. Evaluating a symmetric multilinear form repeatedly on the same vector produces a homogeneous polynomial, and the reverse direction is governed by polarization. Because polarization requires division by factorials, the field characteristic matters. [quotetheorem:9259] This relationship links the concept to multilinear maps, quadratic forms, and higher-order differential data. It also explains a common piece of terminology: when geometers speak about forms, they are usually emphasizing this homogeneous, tensor-like behaviour rather than merely the total degree of an arbitrary polynomial. [remark: Forms] In algebra and geometry, a homogeneous polynomial is often called a form. Thus a linear form has degree $1$, a quadratic form has degree $2$, a cubic form has degree $3$, and a degree-$d$ form is a homogeneous polynomial of degree $d$. [/remark] The word "form" is short, but it carries the same homogeneity requirement. A polynomial such as $x^2+y+1$ is not a quadratic form, even though its total degree is $2$. ## References [Polynomial](/page/Polynomial). [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry). [Vector Space](/page/Vector%20Space). Robin Hartshorne, *Algebraic Geometry* (1977). David Cox, John Little, and Donal O'Shea, *Ideals, Varieties, and Algorithms* (2015).