In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. 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. 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. 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. 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

in any field containing the coefficients of P then the polynomial is homogeneous of degree d.

In particular, if P is homogeneous then

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 is the direct sum (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

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if is a homogeneous polynomial of degree one has, whichever is the commutative ring of the coefficients,

denotes the formal partial derivative of

Homogenization

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

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

then

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

See also

• Multi-homogeneous polynomial

• Quasi-homogeneous polynomial

• 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

Notes

References

External links