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Homogeneous polynomial

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In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. For example, x 5 + 2 x 3 y 2 + 9 x y 4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x 3 + 3 x 2 y + z 7 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. 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.

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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

P ( λ x 1 , … , λ x n ) = λ d P ( x 1 , … , x n ) ,

for every λ in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many λ then the polynomial is homogeneous of degree d.

In particular, if P is homogeneous then

P ( x 1 , … , x n ) = 0 ⇒ P ( λ x 1 , … , λ x n ) = 0 ,

for every λ . 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 R = K [ x 1 , … , 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 R d . The above unique decomposition means that R is the direct sum of the R d (sum over all nonnegative integers).

The dimension of the vector space (or free module) 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

( d + n − 1 n − 1 ) = ( d + n − 1 d ) = ( 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 x 1 , … , x n , one has, whichever is the commutative ring of the coefficients,

d P = ∑ i = 1 n x i ∂ P ∂ x i ,

where ∂ P ∂ x i denotes the formal partial derivative of P with respect to x i .

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:

h P ( x 0 , x 1 , … , x n ) = x 0 d P ( x 1 x 0 , … , x n x 0 ) ,

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

P = x 3 3 + x 1 x 2 + 7 ,

then

h P = x 3 3 + x 0 x 1 x 2 + 7 x 0 3 .

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

P ( x 1 , … , x n ) = h P ( 1 , x 1 , … , x n ) .

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 Kn 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 (x1, ..., xn) in Kn such that f(x1,...,xn) = 0 and at least one of the xi 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.

References

Homogeneous polynomial Wikipedia
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