Homogeneous polynomial
Properties

A homogeneous polynomial defines a homogeneous function. This means that, if a multivariate polynomial P is homogeneous of degree d, then

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

In particular, if P is homogeneous then

for every λ. {\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 R = K {\displaystyle R=K} 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. {\displaystyle R_{d}.} The above unique decomposition means that R {\displaystyle R} is the direct sum of the R d {\displaystyle R_{d}} (sum over all nonnegative integers).

The dimension of the vector space (or free module) R d {\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

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, {\displaystyle x_{1},\ldots,x_{n},} one has, whichever is the commutative ring of the coefficients,

where ∂ P ∂ x i {\displaystyle \textstyle {\frac {\partial P}{\partial x_{i}}}} denotes the formal partial derivative of P with respect to x i. {\displaystyle x_{i}.}
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