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v
(
η
(
ω
p
+
i
)) =
N
+1+
π
p
+
i
,
1
≤
i
≤
m.
(4)
So we have the following theorem.
Theorem 4.
If M
N
(
z
)
is an Nth minimal partial realization of T then
p
deg(det(
M
N
(
z
))) =
−
π
j
.
j
=1
Proof.
Let
ω
1
,...,ω
n
be a normal basis of
Λ
.By(3)wehave
n
v
(
ω
j
)=det(
Λ
)=
−
m
(
N
+1)
.
j
=1
Thus the result follows from (4) and Lemma 1.
As we know, the realization pair (Pol(
T
(
z
)
M
N
(
z
))
,M
N
(
z
)) is defined at best
only up to right multiplication by an
m
m
unimodular matrix. In the following
we first give all solutions of the
N
th right
i
-minimal polynomial column vectors
of
T
for 1
×
≤
i
≤
m
.
Theorem 5.
Let ω
1
,
,ω
n
be a normal basis for the lattice Λ.Thenallthe
Nth right i-minimal polynomial column vectors of T ,
1
···
≤
i
≤
m,areobtained
from
n
η
(
ω
p
+
i
+
f
i,j
(
z
)
ω
j
)
,
j
=1
,j
=
p
+
i
where f
i,j
(
z
)
∈
IF [
z
]
and
deg(
f
i,j
(
z
))
≤
π
p
+
i
−
π
j
with
1
≤
j
≤
n and j
=
p
+
i.
Proof.
Assume
c
(
z
)isan
N
th right
i
-minimal polynomial column vector of
T
for 1
m
.ByTheorem2wehave
η
−
1
(
c
(
z
)) =
γ
≤
i
≤
∈
S
i
(
Λ
)and
v
(
γ
)=
v
(
ω
p
+
i
)
.
(5)
So
γ
canbewrittenastheform
γ
=
j
=1
f
i,j
(
z
)
ω
i
.Since
ω
1
,...,ω
n
are reduced,
θ
(
ω
1
)
,...,θ
(
ω
n
) are linearly independent over IF. By (5), we have
f
i, p
+
i
(
z
)=1
and deg(
f
i,j
(
z
)) +
v
(
ω
j
)=
v
(
ω
p
+
i
) for all
j
,1
=
p
+
i
.Theresult
is easily obtained since
η
restricted on
S
i
(
Λ
) is one-to-one correspondence.
≤
j
≤
n
and
j
In addition we can parameterize all minimal partial realizations as in [5, 13, 18],
that is,
p
M
N
(
z
)=(
η
(
ω
p
+
i
+
f
i,j
(
z
)
ω
j
))
1
≤i≤m
,
j
=1
where
f
i,j
(
z
) is defined as above.
From Theorem 5 we can give the sucient and necessary condition for the
uniqueness of the minimal partial realizations [5, 13, 18].
Theorem 6.
The Nth minimal partial realization of T is unique if and only if
π
p
+
i
<π
1
for i
=1
,...,m.
