- Source: Homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. 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. 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
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:
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
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
Notes
References
External links
Media related to Homogeneous polynomials at Wikimedia Commons
Weisstein, Eric W. "Homogeneous Polynomial". MathWorld.
Kata Kunci Pencarian:
- Homogeneous polynomial
- Homogeneous function
- Discriminant
- Complete homogeneous symmetric polynomial
- Elementary symmetric polynomial
- Quasi-homogeneous polynomial
- Resultant
- Composition (combinatorics)
- Bézout's theorem
- Polynomial
