In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree.
is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.
An algebraic form, or simply form, is a function defined by a homogeneous polynomial.
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.
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.
This means that, if a multivariate polynomial P is homogeneous of degree d, then for every
in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many
This property is fundamental in the definition of a projective variety.
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 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).
That is, if P is a homogeneous polynomial of degree d in the indeterminates
one has, whichever is the commutative ring of the coefficients, where
denotes the formal partial derivative of P with respect to
A non-homogeneous polynomial P(x1,...,xn) can be homogenized by introducing an additional variable x0 and defining the homogeneous polynomial sometimes denoted hP:[2] where d is the degree of P. For example, if then A homogenized polynomial can be dehomogenized by setting the additional variable x0 = 1.