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, x5+2x3y2+9xy4displaystyle x^5+2x^3y^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 x3+3x2y+z7displaystyle x^3+3x^2y+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.

Contents

• 
  1 Properties
  


• 
  2 Homogenization
  


• 
  3 See also
  


• 
  4 References
  


• 
  5 External links
  

Properties

A homogeneous polynomial defines a homogeneous function. This means that, if a multivariate polynomial P is homogeneous of degree d, then

P(λx1,…,λxn)=λdP(x1,…,xn),displaystyle P(lambda x_1,ldots ,lambda x_n)=lambda ^d,P(x_1,ldots ,x_n),,

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

In particular, if P is homogeneous then

P(x1,…,xn)=0⇒P(λx1,…,λxn)=0,displaystyle P(x_1,ldots ,x_n)=0quad 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[x1,…,xn]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 Rd.displaystyle R_d. The above unique decomposition means that Rdisplaystyle R is the direct sum of the Rddisplaystyle R_d (sum over all nonnegative integers).

The dimension of the vector space (or free module) Rddisplaystyle 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−1n−1)=(d+n−1d)=(d+n−1)!d!(n−1)!.displaystyle binom d+n-1n-1=binom d+n-1d=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 x1,…,xn,displaystyle x_1,ldots ,x_n, one has, whichever is the commutative ring of the coefficients,

dP=∑i=1nxi∂P∂xi,displaystyle dP=sum _i=1^nx_ifrac partial Ppartial x_i,

where ∂P∂xidisplaystyle textstyle frac partial Ppartial x_i denotes the formal partial derivative of P with respect to xi.displaystyle x_i.

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]

hP(x0,x1,…,xn)=x0dP(x1x0,…,xnx0),displaystyle ^h!P(x_0,x_1,dots ,x_n)=x_0^dPleft(frac x_1x_0,dots ,frac x_nx_0right),

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

P=x33+x1x2+7,displaystyle P=x_3^3+x_1x_2+7,

then

hP=x33+x0x1x2+7x03.displaystyle ^h!P=x_3^3+x_0x_1x_2+7x_0^3.

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

P(x1,…,xn)=hP(1,x1,…,xn).displaystyle P(x_1,dots ,x_n)=^h!P(1,x_1,dots ,x_n).

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

References

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

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