Homogeneous polynomial

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

${\displaystyle P(\lambda x_1,\dots ,\lambda x_n)={\lambda }^{d}P(x_1,\dots ,x_n),}$

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

In particular, if P is homogeneous then

${\displaystyle P(x_1,\dots ,x_n)=0\Rightarrow P(\lambda x_1,\dots ,\lambda x_n)=0,}$

for every ${\displaystyle \lambda .}$ 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 ${\displaystyle R=K[x_1,\dots ,x_n]}$ over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted ${\displaystyle R_d.}$ The above unique decomposition means that ${\displaystyle R}$ is the direct sum of the ${\displaystyle R_d}$ (sum over all nonnegative integers).

The dimension of the vector space (or free module) ${\displaystyle R_d}$ 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

${\displaystyle (\genfrac{}{}{0ex}{}{d+n-1}{n-1})=(\genfrac{}{}{0ex}{}{d+n-1}{d})=\frac{(d+n-1)!}{d!(n-1)!}.}$

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if P is a homogeneous polynomial of degree d in the indeterminates ${\displaystyle x_1,\dots ,x_n,}$ one has, whichever is the commutative ring of the coefficients,

${\displaystyle dP=\sum _{i=1}^n{x}_i\frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_i},}$

where ${\displaystyle \frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_i}}$ denotes the formal partial derivative of P with respect to ${\displaystyle x_i.}$