Homogeneous polynomial

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Short description: Polynomial whose all nonzero terms have the same degree


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.[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, 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,...,xn) can be homogenized by introducing an additional variable x0 and defining the homogeneous polynomial sometimes denoted hP:[5]

[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_3^3 + x_1 x_2+7, }[/math]

then

[math]\displaystyle{ ^h\!P=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

References

  1. Cox, David A.; Little, John; O'Shea, Donal (2005). Using Algebraic Geometry. Graduate Texts in Mathematics. 185 (2nd ed.). Springer. p. 2. ISBN 978-0-387-20733-9. https://books.google.com/books?id=QFFpepgQgT0C&pg=PP1. 
  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. Cox, Little & O'Shea 2005, p. 35

External links