Graphics Reference
In-Depth Information
Proof.
See [Walk50].
B.7.6. Proposition.
A polynomial f ΠR[X
1
,X
2
,...,X
n
] of degree k is homogeneous
of degree k if and only if
(
)
=
k
(
)
ftX tX
,
,...,
tX
t fX X
,
,...,
X
.
(B.4)
1
2
n
1
2
n
Proof.
The only nontrivial part is showing that if f satisfies equation (B.4), then
it is homogeneous of degree k, but this reduces to the easy case n = 1 by setting
X
2
, X
3
,..., X
n
to X
1
.
Definition.
A polynomial f ΠR[X
1
,X
2
,...,X
n
] is said to be
symmetric
if
(
)
=
(
)
fX X
,
,...,
X
fX
,
X
,...,
X
12
n
s
()
1
s
()
2
s
( )
n
for all permutations s of {1,2, ...,n}.
Consider the equation
n
(
)
(
)
◊◊◊
(
)
=
n
n
-
1
n
-
2
- ◊◊◊-
()
ZXZX
-
-
ZX
-
s
Z
-
s
Z
-
s
Z
1
s ,
(B.5)
1
2
n
0
1
2
n
where the left-hand side is a polynomial in an indeterminate Z over the ring
R[X
1
,X
2
,...,X
n
] and the right-hand side is its expansion, which defines polynomials
s
i
=s
i
(X
1
,X
2
,...,X
n
). Because the left-hand side of equation (B.5) is unchanged by
permuting the X
i
's, the polynomials s
i
are symmetric.
Definition.
The polynomial s
i
is called the
ith elementary symmetric polynomial
in
the variables (indeterminates) X
1
, X
2
,..., X
n
.
For example, if n = 3, then
(
)
=++
s
s
s
XX X
,
,
X X
X
,
1123
1
2
3
(
)
=
X
,
X
,
X
X X
+
X X
+
X X
,
and
2 1 2 3
12
13
23
(
)
=
XX X
,
,
XXX
.
3123
123
The elementary symmetric polynomials form a basis for all symmetric
polynomials.
B.7.7. Theorem.
(The Fundamental Theorem of Symmetric Polynomials) If
f(X
1
,X
2
,...,X
n
) is a symmetric polynomial over a ring R with unity, then there exists
a unique polynomial F(X
1
,X
2
,...,X
n
) over R, so that
(
)
=
(
)
fX X
,
,...,
X
F
ss
,
,...,
s
,
12
n
12
n
where s
i
is the ith elementary polynomial in X
1
, X
2
,..., X
n
.
Proof.
See [Jaco66].
