Homogeneous polynomial

Polynomial whose nonzero terms all 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, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{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 {\displaystyle x^{3}+3x^{2}y+z^{7}} 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

P ( λ x 1 , … , λ x n ) = λ d P ( x 1 , … , x n ) , {\displaystyle P(\lambda x_{1},\ldots ,\lambda x_{n})=\lambda ^{d}\,P(x_{1},\ldots ,x_{n})\,,}

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

In particular, if P is homogeneous then

P ( x 1 , … , x n ) = 0 ⇒ P ( λ x 1 , … , λ x n ) = 0 , {\displaystyle P(x_{1},\ldots ,x_{n})=0\quad \Rightarrow \quad P(\lambda x_{1},\ldots ,\lambda x_{n})=0,}

for every λ . {\displaystyle \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 , … , x n ] {\displaystyle 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 . {\displaystyle R_{d}.} The above unique decomposition means that R {\displaystyle R} is the direct sum of the R d {\displaystyle R_{d}} (sum over all nonnegative integers).

The dimension of the vector space (or free module) R d {\displaystyle 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

( d + n − 1 n − 1 ) = ( d + n − 1 d ) = ( d + n − 1 ) ! d ! ( n − 1 ) ! . {\displaystyle {\binom {d+n-1}{n-1}}={\binom {d+n-1}{d}}={\frac {(d+n-1)!}{d!(n-1)!}}.}

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if P is a homogeneous polynomial of degree d in the indeterminates x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} one has, whichever is the commutative ring of the coefficients,

d P = ∑ i = 1 n x i ∂ P ∂ x i , {\displaystyle dP=\sum _{i=1}^{n}x_{i}{\frac {\partial P}{\partial x_{i}}},}

where ∂ P ∂ x i {\displaystyle \textstyle {\frac {\partial P}{\partial x_{i}}}} denotes the formal partial derivative of P with respect to x i . {\displaystyle x_{i}.}

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:[2]

h P ( x 0 , x 1 , … , x n ) = x 0 d P ( x 1 x 0 , … , x n x 0 ) , {\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),}

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

P ( x 1 , x 2 , x 3 ) = x 3 3 + x 1 x 2 + 7 , {\displaystyle P(x_{1},x_{2},x_{3})=x_{3}^{3}+x_{1}x_{2}+7,}

then

h P ( x 0 , x 1 , x 2 , x 3 ) = x 3 3 + x 0 x 1 x 2 + 7 x 0 3 . {\displaystyle ^{h}\!P(x_{0},x_{1},x_{2},x_{3})=x_{3}^{3}+x_{0}x_{1}x_{2}+7x_{0}^{3}.}

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

P ( x 1 , … , x n ) = h P ( 1 , x 1 , … , x n ) . {\displaystyle P(x_{1},\dots ,x_{n})={^{h}\!P}(1,x_{1},\dots ,x_{n}).}

See also

Notes

  1. ^ 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.
  2. ^ 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.
  3. ^ Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.

References

  1. ^ Cox, David A.; Little, John; O'Shea, Donal (2005). Using Algebraic Geometry. Graduate Texts in Mathematics. Vol. 185 (2nd ed.). Springer. p. 2. ISBN 978-0-387-20733-9.
  2. ^ Cox, Little & O'Shea 2005, p. 35

External links