In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree.^[1] For example, ${\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4))$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial ${\displaystyle x^{3}+3x^{2}y+z^{7))$ is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function.

An algebraic form, or simply form, is a function defined by a homogeneous polynomial.^{[notes 1]} A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.

A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form.^{[notes 2]} A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.

Homogeneous polynomials are ubiquitous in mathematics and physics.^{[notes 3]} They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.

Properties

A homogeneous polynomial defines a homogeneous function. This means that, if a multivariate polynomial P is homogeneous of degree d, then

${\displaystyle P(\lambda x_{1},\ldots ,\lambda x_{n})=\lambda ^{d}\,P(x_{1},\ldots ,x_{n})\,,}$

for every ${\displaystyle \lambda }$ in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many ${\displaystyle \lambda }$ then the polynomial is homogeneous of degree d.

In particular, if P is homogeneous then

${\displaystyle P(x_{1},\ldots ,x_{n})=0\quad \Rightarrow \quad P(\lambda x_{1},\ldots ,\lambda x_{n})=0,}$

for every ${\displaystyle \lambda .}$ This property is fundamental in the definition of a projective variety.

Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial.

Given a polynomial ring ${\displaystyle R=K[x_{1},\ldots ,x_{n}]}$ over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted ${\displaystyle R_{d}.}$ The above unique decomposition means that ${\displaystyle R}$ is the direct sum of the ${\displaystyle R_{d))$ (sum over all nonnegative integers).

The dimension of the vector space (or free module) ${\displaystyle R_{d))$ is the number of different monomials of degree d in n variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree d in n variables). It is equal to the binomial coefficient

${\displaystyle {\binom {d+n-1}{n-1))={\binom {d+n-1}{d))={\frac {(d+n-1)!}{d!(n-1)!)).}$

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if P is a homogeneous polynomial of degree d in the indeterminates ${\displaystyle x_{1},\ldots ,x_{n},}$ one has, whichever is the commutative ring of the coefficients,

${\displaystyle dP=\sum _{i=1}^{n}x_{i}{\frac {\partial P}{\partial x_{i))},}$

where ${\displaystyle \textstyle {\frac {\partial P}{\partial x_{i))))$ denotes the formal partial derivative of P with respect to ${\displaystyle x_{i}.}$

Homogenization

A non-homogeneous polynomial P(x₁,...,x_n) can be homogenized by introducing an additional variable x₀ and defining the homogeneous polynomial sometimes denoted ^hP:^[2]

${\displaystyle {^{h}\!P}(x_{0},x_{1},\dots ,x_{n})=x_{0}^{d}P\left({\frac {x_{1)){x_{0))},\dots ,{\frac {x_{n)){x_{0))}\right),}$

where d is the degree of P. For example, if

${\displaystyle P(x_{1},x_{2},x_{3})=x_{3}^{3}+x_{1}x_{2}+7,}$

then

${\displaystyle ^{h}\!P(x_{0},x_{1},x_{2},x_{3})=x_{3}^{3}+x_{0}x_{1}x_{2}+7x_{0}^{3}.}$

A homogenized polynomial can be dehomogenized by setting the additional variable x₀ = 1. That is

${\displaystyle P(x_{1},\dots ,x_{n})={^{h}\!P}(1,x_{1},\dots ,x_{n}).}$