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, [math]\displaystyle{ x^5 + 2 x^3 y^2 + 9 x y^4 }[/math] is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial [math]\displaystyle{ x^3 + 3 x^2 y + z^7 }[/math] 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

[math]\displaystyle{ P(\lambda x_1, \ldots, \lambda x_n)=\lambda^d\,P(x_1,\ldots,x_n)\,, }[/math]

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

In particular, if P is homogeneous then

[math]\displaystyle{ P(x_1,\ldots,x_n)=0 \quad\Rightarrow\quad P(\lambda x_1, \ldots, \lambda x_n)=0, }[/math]

for every [math]\displaystyle{ \lambda. }[/math] 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 [math]\displaystyle{ R=K[x_1, \ldots,x_n] }[/math] over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted [math]\displaystyle{ R_d. }[/math] The above unique decomposition means that [math]\displaystyle{ R }[/math] is the direct sum of the [math]\displaystyle{ R_d }[/math] (sum over all nonnegative integers).

The dimension of the vector space (or free module) [math]\displaystyle{ R_d }[/math] 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

[math]\displaystyle{ \binom{d+n-1}{n-1}=\binom{d+n-1}{d}=\frac{(d+n-1)!}{d!(n-1)!}. }[/math]

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if P is a homogeneous polynomial of degree d in the indeterminates [math]\displaystyle{ x_1, \ldots, x_n, }[/math] one has, whichever is the commutative ring of the coefficients,

[math]\displaystyle{ dP=\sum_{i=1}^n x_i\frac{\partial P}{\partial x_i}, }[/math]

where [math]\displaystyle{ \textstyle \frac{\partial P}{\partial x_i} }[/math] denotes the formal partial derivative of P with respect to [math]\displaystyle{ x_i. }[/math]

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]

[math]\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 ), }[/math]

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

[math]\displaystyle{ P(x_1,x_2,x_3)=x_3^3 + x_1 x_2+7, }[/math]

then

[math]\displaystyle{ ^h\!P(x_0,x_1,x_2,x_3)=x_3^3 + x_0 x_1x_2 + 7 x_0^3. }[/math]

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

[math]\displaystyle{ P(x_1,\dots, x_n)={^h\!P}(1,x_1,\dots, x_n). }[/math]

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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