Английская Википедия:Homogeneous polynomial

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In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, <math>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>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. 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

<math>P(\lambda x_1, \ldots, \lambda x_n)=\lambda^d\,P(x_1,\ldots,x_n)\,,</math>

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

In particular, if P is homogeneous then

<math>P(x_1,\ldots,x_n)=0 \quad\Rightarrow\quad P(\lambda x_1, \ldots, \lambda x_n)=0,</math>

for every <math>\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>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>R_d.</math> The above unique decomposition means that <math>R</math> is the direct sum of the <math>R_d</math> (sum over all nonnegative integers).

The dimension of the vector space (or free module) <math>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>\binom{d+n-1}{n-1}=\binom{d+n-1}{d}=\frac{(d+n-1)!}{d!(n-1)!}.</math>

Шаблон:Anchor Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if Шаблон:Math is a homogeneous polynomial of degree Шаблон:Math in the indeterminates <math>x_1, \ldots, x_n,</math> one has, whichever is the commutative ring of the coefficients,

<math>dP=\sum_{i=1}^n x_i\frac{\partial P}{\partial x_i},</math>

where <math>\textstyle \frac{\partial P}{\partial x_i}</math> denotes the formal partial derivative of Шаблон:Math with respect to <math>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:

<math>{^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>P(x_1,x_2,x_3)=x_3^3 + x_1 x_2+7,</math>

then

<math>^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>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

External links

• Шаблон:Commonscatinline
• Шаблон:Mathworld