Homogeneous polynomial

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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 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[edit]

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\Longrightarrow\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[edit]

A non-homogeneous polynomial P(x1,...,xn) 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[edit]

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, ..., xn) in Kn such that f(x1,...,xn) = 0 and at least one of the xi 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.

See also[edit]

References[edit]

  1. ^ D. Cox, J. Little, D. O'Shea: Using Algebraic Geometry, 2nd ed., page 2. Springer-Verlag, 2005.
  2. ^ 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.
  3. ^ 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.
  4. ^ Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.
  5. ^ D. Cox, J. Little, D. O'Shea: Using Algebraic Geometry, 2nd ed., page 35. Springer-Verlag, 2005.

External links[edit]