Homogeneous polynomial

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, formula_1 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial formula_2 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.
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
for every formula_4 in any field containing the coefficients of "P". Conversely, if the above relation is true for infinitely many formula_4 then the polynomial is homogeneous of degree "d".
In particular, if "P" is homogeneous then 
for every formula_7 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 formula_8 over a field (or, more generally, a ring) "K", the homogeneous polynomials of degree "d" form
a vector space (or a module), commonly denoted formula_9 The above unique decomposition means that formula_10 is the direct sum of the formula_11 (sum over all nonnegative integers).
The dimension of the vector space (or free module) formula_11 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 in the indeterminates formula_14 one has, whichever is the commutative ring of the coefficients,
where formula_16 denotes the formal partial derivative of with respect to formula_17
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":
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