Graphics Reference
In-Depth Information
()
0
RR
R
=
=
()
(
)
n
n
-
1
[
]
R
X
,
for
n
>
0
.
n
Definition.
R
((n))
is called the
formal power series ring over R in indeterminates
X
1
, X
2
,..., and X
n
and is denoted by R[[X
1
,X
2
,...,X
n
]]. R
(n)
is called the
polynomial
ring over R in indeterminates X
1
, X
2
,..., and X
n
and is denoted by R[X
1
,X
2
,...,X
n
].
Note that with our notation, R[[X
1
,X
2
,...,X
n
]] = R[[X
1
,X
2
,...,X
n-1
]] [[X
n
]] and
similarly for the polynomial ring. From this, Theorem B.7.3, and induction on n we
get
B.7.4. Corollary.
If R is a UFD, then both R[[X
1
,X
2
,...,X
n
]] and R[X
1
,X
2
,...,X
n
]
are UFDs.
Now, every polynomial f = f(X
1
,X
2
,...,X
n
) ΠR[X
1
,X
2
,...,X
n
] can be written as a
finite sum in the form
Â
rr
r
a
X
X
◊◊◊
X
,
where
a
Œ
R
.
(B.3)
12
n
rr
◊◊◊
r
rr
◊◊◊
r
12
12
n
12
n
Definition.
An expression of the form
rr
r
n
aX
12
◊◊◊
X
X
,
12
where a is a nonzero element of R, is called a
monomial
of
total degree
d, where
dr r
=++◊ ◊ ◊ +
12
n
.
The element a is called the
coefficient
of the monomial. If a = 1, we call the expres-
sion a
power product
.
Definition.
Let f ΠR[X
1
,X
2
,...,X
n
]. The
total degree
or simply
degree
of the poly-
nomial f is the largest total degree of all the monomials appearing in f. We say that f
is a
quadratic
,
cubic
,...
polynomial
if its degree is 2, 3, ..., respectively. The polyno-
mial f is said to be
homogeneous of degree d
or simply
homogeneous
if each monomial
that appears in it has total degree d.
Note that any formal power series f in R[[X
1
,X
2
,...,X
n
]] can be written uniquely
in the form
ff f f
=
+
+
+ ◊◊◊
012
where each f
i
is a homogeneous polynomial of degree i.
Definition.
If f
i
is not the zero polynomial, then f
i
is called the ith
homogeneous
component
of f.
Every polynomial f can be written uniquely in the form
