# Homogeneous polynomial

In mathematics, a **homogeneous polynomial** is a polynomial whose nonzero terms all have the same degree.^{[1]} For example, is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines a homogeneous function. An **algebraic form**, or simply **form**, is a function defined by a homogeneous polynomial.^{[2]} 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.^{[3]} A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of quadratic form.

Homogeneous polynomials are ubiquitous in mathematics and physics.^{[4]} 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 a multivariate polynomial *P* is homogeneous of degree *d* if and only if

for every in the field of the coefficients. In particular, if *P* is homogeneous then

for every 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 over a field (or, more generally, a ring) *K*, the homogeneous polynomials of degree *d* form
a vector space (or a module), commonly denoted The above unique decomposition means that is the direct sum of the (sum over all nonnegative integers).

The dimension of the vector space (or free module) 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

## Homogenization

A non-homogeneous polynomial *P*(*x*_{1},...,*x*_{n}) can be homogenized by introducing an additional variable *x*_{0} and defining the homogeneous polynomial sometimes denoted ^{h}*P*:^{[5]}

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

then

A homogenized polynomial can be dehomogenized by setting the additional variable *x*_{0} = 1. That is

## Algebraic forms in general

Algebraic forms, or simply forms, generalize quadratic forms to any degree, and have in the past also been known as *quantics* (a term that originated with Cayley). To specify a type of form, one has to give the degree *d* and the number of variables *n*. A form is *over* some given field *K*, if it maps from *K*^{n} to *K*, where *n* is the number of variables of the form.

A form *f* over some field *K* in *n* variables represents 0 if there exists an element (*x*_{1}, ..., *x _{n}*) in

*K*

^{n}such that

*f*(

*x*

_{1},...,

*x*

_{n}) = 0 and at least one of the

*x*

_{i}is not equal to zero.

A quadratic form over the field of the real numbers represents 0 if and only if it is not definite.

## See also

- Diagonal form
- Graded algebra
- Hilbert series and Hilbert polynomial
- Multilinear form
- Multilinear map
- Polarization of an algebraic form
- Schur polynomial
- Symbol of a differential operator

## References

- ↑ D. Cox, J. Little, D. O'Shea:
*Using Algebraic Geometry*, 2nd ed., page 2. Springer-Verlag, 2005. - ↑ However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms
*homogeneous polynomial*and*form*are sometimes considered as synonymous. - ↑
*Linear forms*are defined only for finite-dimensional vector space, and have thus to be distinguished from*linear functionals*, which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces. - ↑ Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.
- ↑ D. Cox, J. Little, D. O'Shea:
*Using Algebraic Geometry*, 2nd ed., page 35. Springer-Verlag, 2005.