Graphics Reference
In-Depth Information
ff f
=
+
+ ◊◊◊+
d
01
where each f
i
is a homogeneous polynomial of degree i and d is the total degree of f.
Definition.
If f is nonzero, then the smallest integer i, so that f
i
π 0, is called the
order
of f and is denoted by ord(f). The order of 0, ord(0), is defined to be •.
Finally, note that if f ΠR[[X
1
,X
2
,...,X
n
]], then for each i, if we let
[
[
]
]
RRX
=
,...,
X X
,
,...,
X
,
i
1
i
-
1
i
+
1
n
then f ΠR
i
[[X
i
]]. The same holds for the polynomial rings.
Definition.
Let
Â
rr
r
f
=
a
X
X
◊◊◊
X
,
where
a
Œ
R
12
n
rr
◊◊◊
r
rr
◊◊◊
r
12
12
n
12
n
be a polynomial in R[X
1
,X
2,
...,X
n
]. The map
n
RR
p
f
:
Æ
Â
rr
r
(
)
Æ
cc
,
,...,
c
a
cc
12
◊◊◊
c
n
12
n
r r
◊◊◊
r
12
12
n
is called the
evaluation map
associated to f and p
f
(c
1
,c
2
,...,c
n
) will be denoted by
f(c
1
,c
2
,...,c
n
). If
(
)
=
fc c
,
,...,
c
n
0
,
12
then (c
1
,c
2
,...,c
n
) is called a
zero
or
root
of f.
An evaluation map (other than at 0) does not make sense for formal power series
without a notion of limit; however, substitution or composition does as long as the
series we are substituting has no constant term. A precise definition of the formal
power series that is the composition of two power series is actually somewhat tech-
nical (see [ZarS60], Volume II or [Walk50]) and we shall not take the time to do this
here. Intuitively,
Definition.
If f(X), g(X) ΠR[[X]], then the
composition
of f and g, denoted by
(f
g)(X), is the power series where X in f(X) is replaced by g(X) and we collect all the
coefficients of all the same powers of X in the result. The power series f
g is also
referred to as the power series obtained from f by
substitution
of g into f.
We cannot allow g to have a constant term, otherwise the composition would
potentially have a constant term that would be an infinite sum of elements of R.
B.7.5. Theorem.
If f, g ΠR[[X]], where ord(g) > 0, then f
g is well defined.
Furthermore,
(1) If fg π 0, then ord(f
g) = ord(f) ord(g).
(2) If h ΠR[[X]] and ord(h) > 0, then f
(g
h) = (f
g)
h.
