Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.^[1] For example, $x^5 + 2 x^3 y^2 + 9 x y^4$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial $x^3 + 3 x^2 y + z^7$ 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.^[2] 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.^[3] 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.^[4] 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 a multivariate polynomial P is homogeneous of degree d if and only if

P(\lambda x_1, \ldots, \lambda x_n)=\lambda^d\,P(x_1,\ldots,x_n)\,,

for every $\lambda$ in the field of the coefficients. 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)!}.

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:^[5]

{^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_1x_2 + 7 x_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).

Algebraic forms in general

Algebraic forms, or simply forms, generalize quadratic forms to any degree, and have in the past also been known as quantics (a term that originated with Cayley). To specify a type of form, one has to give the degree d and the number of variables n. A form is over some given field K, if it maps from Kⁿ to K, where n is the number of variables of the form.

A form f over some field K in n variables represents 0 if there exists an element (x₁, ..., x_n) in Kⁿ such that f(x₁,...,x_n) = 0 and at least one of the x_i is not equal to zero.

A quadratic form over the field of the real numbers represents 0 if and only if it is not definite.

References

↑ D. Cox, J. Little, D. O'Shea: Using Algebraic Geometry, 2nd ed., page 2. Springer-Verlag, 2005.
↑ However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms homogeneous polynomial and form are sometimes considered as synonymous.
↑ Linear forms are defined only for finite-dimensional vector space, and have thus to be distinguished from linear functionals, which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces.
↑ Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.
↑ D. Cox, J. Little, D. O'Shea: Using Algebraic Geometry, 2nd ed., page 35. Springer-Verlag, 2005.

External links

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

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