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 coefficientsofP. 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 (orfree module) ${\displaystyle R_{d}}$  is the number of different monomials of degree dinn variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree dinn 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 derivativeofP 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:^[2]

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

where d is the degreeofP. For example, if

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

then

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

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.