Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. For example, $x^{5}+2x^{3}y^{2}+9xy^{4}$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial $x^{3}+3x^{2}y+z^{7}$ is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines 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,...,x_n) 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

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

then

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

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

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

See also

• 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

External links

• Weisstein, Eric W. "Homogeneous Polynomial". MathWorld.