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For any element
,
we denote by
the multidegree of
. That means,
where we employ the lexicographic order on
.
For any subalgebra
of
, we define its multidegree monoid
as follows.
It is a sub monoid of
.
We use several norms for indices in
.
Among them is the ``
-norm''
.
of
The total degree
of an element
of
is then defined to be the
-norm of
.
By a total degree of a derivation or an algebra endomorphism of
we mean the maximum of the total degree of the image of the standard generators
.
Definition 2
For any subset

of

, we denote by

the set
of all elements of

whose

-norms are less than or equal to

.
We denote by

"the affine Hilbert function" of

, that is,
For any subalgebra

of

, we define its affine Hilbert function

as the affine Hilbert function

of multidegree of

.
Lemma 18
Let

be a positive integer.
Let

be a sub monoid of

.
Let

be the submodule of

generated by

.
Then the following conditions are equivalent.
- There exists a positive real number
such that
for all
.
-
:
Take
which are linearly independent over
.
Put
.
Then a map
defined by
is injective and
satisfies
for every positive integer
.
Thus we have
when
is large enough.
(
denotes the Gaussian symbol.)
: Assume on the contrary that
.
Then the module
, being torsion free, is isomorphic to
.
Let
be elements of
which forms a
-basis of
. Then a map
is an injective linear map from
to
.
Thus we may easily see that there exists a real number
such that
This implies that
is smaller than the number of elements
of
which are shorter (in
-norm) than
.
ARRAY(0x8f2aaf8)
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2003/3/3