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, $$x^5 + 2 x^3 y^2 + 9 x y^4$$ 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. 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
 * $$P(\lambda x_1, \ldots, \lambda x_n)=\lambda^d\,P(x_1,\ldots,x_n)\,,$$

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

In particular, if P is homogeneous then
 * $$P(x_1,\ldots,x_n)=0 \quad\Rightarrow\quad P(\lambda x_1, \ldots, \lambda x_n)=0,$$

for every $$\lambda.$$ 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 $$R=K[x_1, \ldots,x_n]$$ 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


 * $$\binom{d+n-1}{n-1}=\binom{d+n-1}{d}=\frac{(d+n-1)!}{d!(n-1)!}.$$

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if $P$ is a homogeneous polynomial of degree $d$ in the indeterminates $$x_1, \ldots, x_n,$$ one has, whichever is the commutative ring of the coefficients,
 * $$dP=\sum_{i=1}^n x_i\frac{\partial P}{\partial x_i},$$

where $$\textstyle \frac{\partial P}{\partial x_i}$$ 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:
 * $${^h\!P}(x_0,x_1,\dots, x_n) = x_0^d P \left (\frac{x_1}{x_0},\dots, \frac{x_n}{x_0} \right ),$$

where d is the degree of P. For example, if
 * $$P(x_1,x_2,x_3)=x_3^3 + x_1 x_2+7,$$

then
 * $$^h\!P(x_0,x_1,x_2,x_3)=x_3^3 + x_0 x_1x_2 + 7 x_0^3.$$

A homogenized polynomial can be dehomogenized by setting the additional variable x0 = 1. That is
 * $$P(x_1,\dots, x_n)={^h\!P}(1,x_1,\dots, x_n).$$