A (multivariable) polynomial is called homogeneous if all monomials that make up the polynomial have the same degree . Homogeneous polynomials are also known as shapes .
definition
Let be a commutative ring with one and the polynomial ring over in indefinite. A monomial is then a polynomial for which one with
R.
{\ displaystyle R}
R.
[
X
1
,
...
,
X
n
]
{\ displaystyle R [X_ {1}, \ dotsc, X_ {n}]}
R.
{\ displaystyle R}
n
{\ displaystyle n}
p
∈
R.
[
X
1
,
...
,
X
n
]
{\ displaystyle p \ in R [X_ {1}, \ dotsc, X_ {n}]}
α
∈
R.
{\ displaystyle \ alpha \ in R}
p
=
α
X
1
i
1
⋅
⋯
⋅
X
n
i
n
{\ displaystyle p = \ alpha X_ {1} ^ {i_ {1}} \ cdot \ dots \ cdot X_ {n} ^ {i_ {n}}}
exists. The degree of this monomial is
d
e
G
(
p
)
=
i
1
+
⋯
+
i
n
.
{\ displaystyle \ mathrm {deg} (p) = i_ {1} + \ dotsb + i_ {n}.}
A polynomial in is called homogeneous if it is a sum of monomials of equal degree.
R.
[
X
1
,
...
,
X
n
]
{\ displaystyle R [X_ {1}, \ dotsc, X_ {n}]}
properties
f
∈
R.
[
X
1
,
...
,
X
n
]
{\ displaystyle f \ in R [X_ {1}, \ dotsc, X_ {n}]}
is homogeneous of degree if and only if :
k
{\ displaystyle k}
R.
[
X
1
,
...
,
X
n
]
[
T
]
{\ displaystyle R [X_ {1}, \ dotsc, X_ {n}] [T]}
f
(
T
X
1
,
...
,
T
X
n
)
=
T
k
⋅
f
(
X
1
,
...
,
X
n
)
{\ displaystyle f (TX_ {1}, \ dotsc, TX_ {n}) = T ^ {k} \ cdot f (X_ {1}, \ dotsc, X_ {n})}
With a polynomial ring over an integrity ring, a product of polynomials is homogeneous if and only if every factor is homogeneous.
Examples
Every monomial is homogeneous.
The set of all homogeneous polynomials in , the polynomial ring in a variable above , is given by
R.
[
X
]
{\ displaystyle R [X]}
R.
{\ displaystyle R}
{
a
X
n
∣
a
∈
R.
,
n
∈
N
∪
{
0
}
}
.
{\ displaystyle \ {aX ^ {n} \; \ mid \; a \ in R, \; n \ in \ mathbb {N} \ cup \ {0 \} \}.}
Simple examples of homogeneous polynomials in (see whole numbers ):
Z
[
X
,
Y
]
{\ displaystyle \ mathbb {Z} [X, Y]}
X
4th
-
Y
4th
{\ displaystyle X ^ {4} -Y ^ {4}}
is homogeneous because of
deg
(
X
4th
)
=
deg
(
Y
4th
)
=
4th
{\ displaystyle \ deg (X ^ {4}) = \ deg (Y ^ {4}) = 4.}
X
7th
+
5
X
3
Y
4th
+
X
Y
6th
{\ displaystyle X ^ {7} + 5X ^ {3} Y ^ {4} + XY ^ {6}}
is homogeneous because of
deg
(
X
7th
)
=
deg
(
X
3
Y
4th
)
=
deg
(
X
Y
6th
)
=
7th
{\ displaystyle \ deg (X ^ {7}) = \ deg (X ^ {3} Y ^ {4}) = \ deg (XY ^ {6}) = 7.}
Examples of non-homogeneous polynomials in (see rational numbers ):
Q
[
X
,
Y
,
Z
]
{\ displaystyle \ mathbb {Q} [X, Y, Z]}
X
4th
Z
-
3
4th
Y
Z
2
{\ displaystyle X ^ {4} Z - {\ frac {3} {4}} YZ ^ {2}}
is not homogeneous because of
deg
(
X
4th
Z
)
=
5
≠
3
=
deg
(
Y
Z
2
)
.
{\ displaystyle \ deg (X ^ {4} Z) = 5 \ neq 3 = \ deg (YZ ^ {2}).}
X
3
Y
3
Z
2
-
3
X
2
Y
6th
-
7th
3
Y
5
{\ displaystyle X ^ {3} Y ^ {3} Z ^ {2} -3X ^ {2} Y ^ {6} - {\ frac {7} {3}} Y ^ {5}}
is not homogeneous because of and
deg
(
X
3
Y
3
Z
2
)
=
deg
(
X
2
Y
6th
)
=
8th
{\ displaystyle \ deg (X ^ {3} Y ^ {3} Z ^ {2}) = \ deg (X ^ {2} Y ^ {6}) = 8}
deg
(
Y
5
)
=
5.
{\ displaystyle \ deg (Y ^ {5}) = 5.}
graduation
Every polynomial can be written in a unique way as the sum of homogeneous polynomials of different degrees by combining all monomials of the same degree. The polynomial ring can thus be written as a direct sum :
R.
[
X
1
,
...
,
X
n
]
=
⨁
d
≥
0
A.
d
,
{\ displaystyle R [X_ {1}, \ dotsc, X_ {n}] = \ bigoplus _ {d \ geq 0} A_ {d},}
in which
A.
d
=
⨁
e
1
+
⋯
+
e
n
=
d
,
e
i
≥
0
R.
⋅
X
1
e
1
⋅
⋯
⋅
X
n
e
n
{\ displaystyle A_ {d} = \ bigoplus _ {e_ {1} + \ dotsb + e_ {n} = d, \ e_ {i} \ geq 0} R \ cdot X_ {1} ^ {e_ {1}} \ cdot \ dots \ cdot X_ {n} ^ {e_ {n}}}
is the set of homogeneous polynomials of degree together with the zero polynomial. It applies
d
{\ displaystyle d}
A.
d
⋅
A.
d
′
⊆
A.
d
+
d
′
,
{\ displaystyle A_ {d} \ cdot A_ {d '} \ subseteq A_ {d + d'},}
the polynomial ring is thus a graduated ring .
generalization
Generally called in a graduated ring
⨁
d
≥
0
A.
d
{\ displaystyle \ bigoplus _ {d \ geq 0} A_ {d}}
the elements out homogeneous by degree .
A.
d
{\ displaystyle A_ {d}}
d
{\ displaystyle d}
See also
Individual evidence
^ Fischer: Textbook of Algebra. 2013, p. 169, Lemma.
^ Fischer: Textbook of Algebra. 2013, p. 169, product of homogeneous polynomials.
literature
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