Homogeneous polynomial

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

${\displaystyle P(\lambda x_{1},\ldots ,\lambda x_{n})=\lambda ^{d}\,P(x_{1},\ldots ,x_{n})\,,}$

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

${\displaystyle P(x_{1},\ldots ,x_{n})=0\quad \Rightarrow \quad P(\lambda x_{1},\ldots ,\lambda x_{n})=0,}$

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 ${\displaystyle R=K[x_{1},\ldots ,x_{n}]}$ over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted ${\displaystyle R_{d}.}$ 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

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

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if P is a homogeneous polynomial of degree d in the indeterminates ${\displaystyle x_{1},\ldots ,x_{n},}$ one has, whichever is the commutative ring of the coefficients,

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

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

A non-homogeneous polynomial P(x₁,...,x_n) can be homogenized by introducing an additional variable x₀ and defining the homogeneous polynomial sometimes denoted ^hP:[5]

${\displaystyle {^{h}\!P}(x_{0},x_{1},\dots ,x_{n})=x_{0}^{d}P\left({\frac {x_{1}}{x_{0}}},\dots ,{\frac {x_{n}}{x_{0}}}\right),}$

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

${\displaystyle P=x_{3}^{3}+x_{1}x_{2}+7,}$

then

${\displaystyle ^{h}\!P=x_{3}^{3}+x_{0}x_{1}x_{2}+7x_{0}^{3}.}$

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

${\displaystyle P(x_{1},\dots ,x_{n})={^{h}\!P}(1,x_{1},\dots ,x_{n}).}$

• Multi-homogeneous polynomial
• Quasi-homogeneous polynomial
• Diagonal form
• Graded algebra
• Hilbert series and Hilbert polynomial
• Multilinear form
• Multilinear map
• Polarization of an algebraic form
• Schur polynomial
• Symbol of a differential operator

• Media related to Homogeneous polynomials at Wikimedia Commons
• Weisstein, Eric W. "Homogeneous Polynomial". MathWorld.