tóm tắt luận án tiến sĩ về tập xác định duy nhất cho hàm chỉnh hình nhiều biến

C
f, g C
f =
ag + b
cg + d
a, b, c, d
ad − bc = 0
f, g C
f ≡ g.
p
p
f, g : C
m
−→ P
n
(C) 3n + 1
P
n
(C) L
P
n
(C) L(f) = g
p
f, g p
a
1
, a
2
, a
3

, a
4
f(z) = a
i
g(z) = a
i
, i = 1, 2, 3, 4 f ≡ g.
A
p
f, g : C
p
−→ P
n
(C
p
)
H
i
, 1  i  3n + 1
P
n
(C
p
) f
−1
(H
i
)

f

−1
(H
j
) = ∅ i = j
f
−1
(H
i
) = g
−1
(H
i
) i = 1, , 3n + 1 f(z) = g(z)
z ∈
3n+1

i=1
f
−1
(H
i
). f ≡ g.
f C ∪ {∞}
E
m
0
f
(S) =

a∈S


(z, m) ∈ C N : f(z) = a m = (m
0
, n)

.
m
0
= ∞( m
0
= 1)
E
∞
f
(S) = E
f
(S), ( E
1
f
(S) = E
f
(S)
P
n
(C) n C
f f = [f
1
: : f
n+1
] : C −→ P

n
(C)
f
1
, , f
n+1

f = (f
1
, , f
n+1
) : C −→ C
n+1
− {0} f

f = (f
1
, , f
n+1
) g = (g
1
, , g
n+1
) f
c f
i
= cg
i
i
f(z) = [c

1
: : c
n+1
] c
1
, , c
n+1
0 f
H P
n
(C)
F = 0, f H
E
f
(H) = E
F
◦

f
(0), E
f
(H) = E
F
◦

f
(0),
E
f
(H,  k) =


z ∈ C : F
◦
˜
f(z) = 0 , v
F
◦
˜
f
(z)  k

.
C ∪ {∞}
S C ∪ {∞}
f, g E
f
(S) = E
g
(S), f ≡ g
{S
1
, S
2
} C ∪ {∞}
f, g E
f
(S
i
) = E
g

(S
i
), i = 1, 2, f ≡ g
S
S
1
, S
2
S = {z ∈ C : z + e
z
= 0},
p
({z
1
, , z
n
}, w)
n ≥ 5 ({z
1
, , z
n
}, w)
n ≥ 4
H
1
, , H
q
P
n
k

1
, , k
q
A
C P
n
(C), f, g ∈ A,
1)E
f
(H
i
,  k
i
) = E
g
(H
i
,  k
i
),
2)f = g
q

i=1
E
f
(H
i
,  k
i

) i = 1, , q g ∈ A.
q k
i
n #A = 1
X P
n
(C) E
f
(X) = E
g
(X)
f ≡ g
p
p
q
k
nd
d ≥ (2s − 1)
2
p k
p
p
p
q
n + 1
p
p
p
p
p

p
f

f f
E
f
(H), E
f
(H), E
f
(H,  k)
f C
a ∈ C, v
f
(a) f a,
f(z) = (z − a)
v
f
(a)
g(z),
g(z) a g(a) = 0
k, l r > 1
v
k
f
: C −→ N v
k
f
(z) =


v
f
(z) v
f
(z)  k,
0 v
f
(z) > k.
n
k
f
(r) =

|z|r
v
k
f
(z), n
k
f
(a, r) = n
k
f−a
(r).
N
k
f
(a, r) =
r


1
n
k
f
(a, x)
x
dx, N
k
f
(r) = N
k
f
(0, r),
N
k
l,f
(a, r) =
r

1
n
k
l,f
(a, x)dx
x
, n
k
l,f
(a, r) =


|z|r
min

v
k
f−a
(z), l

.
v
>k
f
: C −→ N v
>k
f
(z) =

v
f
(z) v
f
(z) > k
0 v
f
(z)  k
n
>k
f
(r) =


|z|r
v
>k
f
(z), n
>k
f
(a, r) = n
>k
f−a
(r),
N
>k
f
(a, r) =
r

1
n
>k
f
(a, x)
x
dx, N
>k
f
(r) = N
>k
f
(0, r),

N
>k
l,f
(a, r) =
r

1
n
>k
l,f
(a, r)
x
dx, n
>k
l,f
(a, r) =

|z|r
min

v
>k
f−a
(z), l

.
X
1
, , X
q

P
n
(C)
n + 1 {X
1
, , X
q
}
f : C −→ P
n
(C)
P F z
1
, . . . , z
n+1
P (

f) = 0
F (

f) = 0).
f m
m
P
n
(C) f m
f : C −→ P
n
(C)


f = (f
1
, , f
n+1
)
T
f
(r) =
1
2π
2π

0
log ||

f(re
ıθ
)||dθ − log ||

f(0)||,
||

f|| =

|f
1
|
2
+ + |f
n+1

|
2

1/2
f, g : C → P
n
(C)
k
1
, . . . , k
q
∈ N
∗
H
1
, . . . , H
q
P
n
(C)
f(C) ⊂ H
i
g(C) ⊂ H
i
i = 1, . . . , q
f(z) = g(z) z ∈
q

i=1
E

f
(H
i
,  k
i
) z ∈
q

i=1
E
g
(H
i
,  k
i
).
q > 2n
2
+ n + 1 +
q

i=1
n
k
i
+ 1
, f ≡ g
E
f
(H

i
,  k
i
)

E
f
(H
j
,  k
j
) = ∅
1  i = j  q
f, g : C −→ P
n
(C)
k
1
, , k
q
∈ N
∗
H
1
, , H
q
P
n
(C)
f(C) ⊂ H

i
g(C) ⊂ H
i
i = 1, . . . , q
E
f
(H
i
,  k
i
)

E
f
(H
j
,  k
j
) = ∅ 1  i = j  q
f(z) = g(z) z ∈ E
f
(H
i
,  k
i
) z ∈ E
g
(H
i
,  k

i
), i = 1, , q.
q > 3n + 1 +
q

i=1
n
k
i
+ 1
f ≡ g
k
i
→ ∞ n = 1 k
i
→ ∞
P ∈ C[z]
f, g
C P (f) = P (g) f = g
P ∈ C[z]
f, g
C c = 0 P (f ) = cP (g), f = g
P z
1
, . . . , z
n+1
f, g : C −→ P
n
(C)


f g P (

f) = P (g) f = g
P z
1
, . . . , z
n+1
f, g : C −→ P
n
(C)

f, g, c = 0
P (

f) = cP (g) f = g
P
n
(C)
x
d
1
+ + x
d
n+1
= 0
f, g : C −→ P
n
(C)
X d, H
1

, , H
q
P
n
(C)
E
f
(X) = E
g
(X),
f(z) = g(z) z ∈
q

i=1
E
f
(H
i
,  k
i
).
q > 2n
2
+ n + 1 +
q

i=1
n
k
i

+ 1
d ≥ (2n + 1)
2
f ≡ g.
E
f
(H
i
,  k
i
)

E
f
(H
j
,  k
j
) = ∅
1  i = j  q
f, g : C −→ P
n
(C)
X d, H
1
, , H
q
P
n
(C)

E
f
(H
i
,  k
i
)

E
f
(H
j
,  k
j
) = ∅ i = j,
E
f
(X) = E
g
(X),
f(z) = g(z) z ∈
q

i=1
E
f
(H
i
,  k
i

).
q > 3n + 1 +
q

i=1
n
k
i
+ 1
d ≥ (2n + 1)
2
f ≡ g.
n ∈ N
∗
n ≥ 2m + 9 (m, n) = 1 m ≥ 2, P
i
(z) = z
n
− a
i
z
n−m
+ b
i
0 = a
i
, b
i
∈ C, i = 1, 2, , s b
2d

i
= b
d
j
b
d
l
i = j, i = l.
Q
i
=

P
i
(z
i
, z
s+1
) = z
n
i
− a
i
z
n−m
i
z
m
s+1
+ b

i
z
n
s+1
, i = 1, 2, , s.
P
s+1,d
= Q
d
1
+ Q
d
2
+ + Q
d
s
, d ≥ (2s − 1)
2
.
P
s+1,d
nd C.
P
s+1,d
,
f, g : C −→ P
s
(C)
X P
s

(C)
P
s+1,d
= 0 E
f
(X) = E
g
(X), f ≡ g.
n ∈ N
∗
, n ≥ 2m + 9, m ≥ 2, (m, n) = 1 P (z) =
z
n
− az
n−m
+ b 0 = a, b ∈ C

P
i
(z
i
, z
j
) = z
n
i
− a
i
z
n−m

i
z
m
j
+ b
i
z
n
j
.
R
1
(z
1
, z
2
) =

P (z
1
, z
2
) = z
n
1
− az
n−m
1
z
m

2
+ bz
n
2
,
R
i
(z
1
, . . . , z
i+1
) = R
1

R
i−1
(z
1
, . . . , z
i
),

P
n
(i−2)
(z
i
, z
i+1
)


, i = 2, , s.
R
s
n
s
C
Y
C P
s
(C).
f, g : C −→ P
s
(C)
Y P
s
(C) R
s
= 0
E
f
(Y ) = E
g
(Y ) f ≡ g.
p
p
p
p
v
f

(a), v
k
f
, v
>k
f

f(z) D
r
H
f
(r) = |f|
r
.
k, l ∈ N
∗
ρ 0 < ρ  r
1)N
k
f
(a, r) =
1
ln p
r

ρ
n
k
f
(a, x)

x
dx, N
k
l,f
(a, r) =
1
ln p
r

ρ
n
k
l,f
(a, x)dx
x
.
2)N
>k
f
(a, r) =
1
ln p
r

ρ
n
>k
f
(a, x)
x

dx, N
>k
l,f
(a, r) =
1
ln p
r

ρ
n
>k
l,f
(a, r)
x
dx.
p
f = [f
1
: : f
n+1
] : C
p
−→ P
n
(C
p
)

f = (f
1

, , f
n+1
) : C
p
−→ C
n+1
p
− {0}

f = (f
1
, , f
n+1
) g = (g
1
, , g
n+1
) f
c f
i
= cg
i
i
f : C
p
−→ P
n
(C
p
)


f = (f
1
, , f
n+1
) f
H
f
(r) = max
1in+1
H
f
i
(r).
p
f, g : C
p
−→ P
n
(C
p
)
k
1
, . . . , k
q
∈ N
∗
H
1

, . . . , H
q
P
n
(C
p
)
f(C
p
) ⊂ H
i
, g(C
p
) ⊂ H
i
i = 1, . . . , q
f(z) = g(z) z ∈
q

i=1
E
f
(H
i
,  k
i
) z ∈
q

i=1

E
g
(H
i
,  k
i
).
q  2n
2
+ n + 1 +
q

i=1
n
k
i
+ 1
f ≡ g
E
f
(H
i
,  k
i
)

E
f
(H
j

,  k
j
) = ∅
1  i = j  q
f, g : C
p
−→ P
n
(C
p
)
k
1
, , k
q
∈ N
∗
H
1
, , H
q
P
n
(C
p
)
f(C
p
) ⊂ H
i

g(C
p
) ⊂ H
i
i = 1, . . . , q
E
f
(H
i
,  k
i
)

E
f
(H
j
,  k
j
) = ∅ 1  i = j  q,
f(z) = g(z) z ∈ E
f
(H
i
,  k
i
) z ∈ E
g
(H
i

,  k
i
), i = 1, , q.
q ≥ 3n + 1 +
q

i=1
n
k
i
+ 1
f ≡ g
k
i
→ ∞
p q
n + 1
f, g : C
p
−→ P
n
(C
p
)
X
i
d H
j
P
n

(C
p
) f g X
i
, H
j
i = 1, . . . , n + 1 j = 1, . . . , q
E
f
(X
i
) = E
g
(X
i
), i = 1 . . . , n + 1,
f(z) = g(z) z ∈
q

j=1
E
f
(H
j
,  k
j
).
q  2n
2
+ n + 1 +

q

i=1
n
k
i
+ 1
f ≡ g.
E
f
(H
i
,  k
i
)

E
f
(H
j
,  k
j
) = ∅
1  i = j  q
f, g : C
p
−→ P
n
(C
p

)
X
i
d H
j
P
n
(C
p
) f g X
i
, H
j
i = 1, . . . , n + 1, j = 1, . . . , q
E
f
(H
i
,  k
i
)

E
f
(H
j
,  k
j
) = ∅ i = j,
E

f
(X
i
) = E
g
(X
i
), i = 1, . . . , n + 1,
f(z) = g(z) z ∈
q

i=1
E
f
(H
i
,  k
i
).
q  3n + 1 +
q

i=1
n
k
i
+ 1
f ≡ g.
p
p

n ∈ N
∗
, n ≥ 2m + 8, m ≥ 2, (m, n) = 1, 0 = a, b ∈ C
p
P (z) = z
n
− az
n−m
+ b
a
n
b
m
=
n
n
m
m
(n − m)
n−m
.

P
i
(z
i
, z
j
) = z
n

i
− az
n−m
i
z
m
j
+ bz
n
j
.
P
i
P
1
(z
1
, z
2
) =

P (z
1
, z
2
) = z
n
1
− az
n−m

1
z
m
2
+ bz
n
2
,
P
i
(z
1
, . . . , z
i+1
) = P
i−1


P (z
1
, z
2
), . . . ,

P (z
i
, z
i+1
)


, i = 2, , s.
P
s
n
s
C
p
P
s
f, g : C
p
−→ P
s
(C
p
)
X P
s
(C
p
)
P
s
= 0 E
f
(X) = E
g
(X), f ≡ g.
p
P (z),


P
i
(z
i
, z
j
)
A
1
(z
1
, z
2
) =

P (z
1
, z
2
),
A
i
(z
1
, , z
i+1
) = A
1


A
i−1
(z
1
, , z
i
),

P
n
(i−2)
(z
i
, z
i+1
)

, i = 2, , s.
A
s
n
s
C
p
Y
C
p
P
s
(C

p
).
f, g : C
p
−→ P
s
(C
p
)
Y P
s
(C
p
)
A
s
= 0 E
f
(Y ) = E
g
(Y ), f ≡ g.
p
p
p
p
p
({a
1
, , a
q

}, {u}) q ≥ 4
C
m
p
p− m
C
m
p
=

(z
1
, , z
m
) : z
i
∈ C
p
i = 1, , m

,
f C
m
p
f =

|γ|≥0
a
γ
z

γ
.
f(z
(m)
)
H
f
(r
(m)
) = log |f |
r
(m)
.
v
d
f
: C
m
p
→ (N

{+∞})
m
v
d
f
(a
(m)
) = (v
1,f−d

(a
(m)
), . . . , v
m,f−d
(a
(m)
)).
ρ
1
, . . . , ρ
m
0 < ρ
i
 r
i
, i = 1, . . . , m.
x ∈ R,
A
i
(x) = (ρ
1
, . . . , ρ
i−1
, x, r
i+1
, . . . , r
m
), i = 1, . . . , m,
B
i

(x) = (ρ
1
, . . . , ρ
i−1
, x, ρ
i+1
, . . . , ρ
m
), i = 1, . . . , m.
N
f
(a, r
(m)
)
N
f
(a, r
(m)
) =
1
ln p
m

i=1
r
i

ρ
i
n

i,f
(a, A
i
(x))
x
dx, n
i,f
(a, r
(m)
) = n
1i,f−a
(r
(m)
).
p
f =
f
1
f
2
C
m
p
, f
1
, f
2
C
m
p

. f
H
f
(r
(m)
) = max
1i2
H
f
i
(r
(m)
).
d ∈ C
p
v
d
f
: C
m
p
→ (N ∪ {+∞})
m
v
d
f
(a
(m)
) = v
0

f
1
−df
2
(a
(m)
), v
∞
f
(a
(m)
) = v
0
f
2
(a
(m)
).
S C
p
∪ {∞} i = 1, 2 , m.
E
i,f
(S) =

d∈S

(q
i
, a

(m)
) ∈ (N ∪ {+∞}) × C
m
p
|f(a
(m)
) = d, v
d
i,f
(a
(m)
) = q
i

,
S = {S
1
, , S
n
} C
p
∪ {∞}
n
C
m
p
f g
C
m
p

E
i,f
(S
j
) = E
i,g
(S
j
),
E
f
(S
j
) = E
g
(S
j
) i = 1, . . . , m, j = 1, n, f ≡ g.
n
n n
1 1
2 2 bi bi
C
p
C
m
p
S C
p
∪{∞}

C
m
p
C
p
S n
C
m
p
n
C
p
f, g C
m
p
E
f
(a
i
) = E
g
(a
i
), a
i
∈ C
p
∪ {∞}, i = 1, 2, , q. q ≥ 4 f ≡ g.
f, g C
m

p
E
i,f
(a
j
) = E
i,g
(a
j
), i = 1 , 2, , m, a
j
∈ C
p
∪ {∞}, j = 1, 2, 3.
f ≡ g.
p
P (x) ∈ C
p
[x]
C
m
p
f, g C
m
p
P (f) = P (g) f = g.
P (x) ∈ C
p
[x]
C

m
p
f, g
C
m
p
c ∈ C
p
P (f) = P (g) f = g
P (x) q
P

(x) = a(x − d
1
)
q
1
. . . (x − d
k
)
q
k
,
q
1
+ · · · + q
k
= q − 1 d
1
, . . . , d

k
P

k P
d
1
, . . . , d
k
∈ C
p
\{0}.
P (x)
(H) P (d
l
) = P (d
m
) 1  l < m  k.
P (x)
(G)
k

i=1
P (d
i
) = 0.
C
p
C
m
p

P (x) ∈ C
p
[x]
C
p
C
m
p
.
P (x) ∈ C
p
[x]
k ≥ 3 (H) P (x)
C
m
p
P (x) ∈ C
p
[x]
k ≥ 3 (H) (G) P(x)
C
m
p
P (x) ∈ C
p
[x]
k ≥ 3 (H) (G)
{a
1
, , a

q
} P (x) = 0

{a
1
, , a
q
}, {∞}

C
m
p
a
i
= u ∈ C
p
1
a
i
− u
, i = 1, , q P (x) = 0

{a
1
, , a
q
}, {u}

C
m

p
q ≥ 4 w ∈ C
p
p

{a
1
, , a
q
}, {u}

p
p
f C
m
p
a
j
∈ C
p
,
j = 1, . . . , q
(q − 1)H
f
(B
e
(r
e
))


q

j=1
N
f
(a
j
, B
e
(r
e
)) + N
f
(∞, B
e
(r
e
)) − N
0,∂
γ
1
f
(B
e
(r
e
)) − log r
e
+ O(1),
O(1) r

e
p
p
p
p
p