Homogeneous polynomial

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree.^[1] For example, ${\displaystyle x^5+2x^3{y}^2+9x{y}^4}$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial ${\displaystyle x^3+3x^{2}y+z^7}$ is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function.

An algebraic form, or simply form, is a function defined by a homogeneous polynomial.^{[notes 1]} 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.^{[notes 2]} 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.^{[notes 3]} 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,

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

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

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:^[2]

${\displaystyle {}^{h}P(x_0,x_1,\dots ,x_n)=x_0^{d}P(\frac{x_1}{x_0},\dots ,\frac{x_n}{x_0}),}$

where d is the degree of P. 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

${\displaystyle P(x_1,\dots ,x_n)={}^{h}P(1,x_1,\dots ,x_n).}$

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

Notes

References

External links

• Weisstein, Eric W.. "Homogeneous Polynomial". http://mathworld.wolfram.com/HomogeneousPolynomial.html. 

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