设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
PureMathematics
n
Ø
ê
Æ
,2022,12(1),14-19
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.121002
Ä
u
ÿ
À
Ý
“
ê
Ä
½
n
y
²
ÆÆÆ
ZZZ
ì
À
à
’
Œ
Æ
§
&
E
‰
Æ
†
ó
§
Æ
§
ì
À
S
Â
v
F
Ï
µ
2021
c
11
28
F
¶
¹
^
F
Ï
µ
2022
c
1
3
F
¶
u
Ù
F
Ï
µ
2022
c
1
10
F
Á
‡
©
^
ÿ
À
Ý
n
Ø
‰
Ñ
“
ê
Ä
½
n
y
²
"
Ä
k
E
Ó
Ô
•
§
§
r
E
,
¯
K
z
¤
˜
‡
{
ü
¯
K
¶
Ï
L
é
Ó
Ô
•
§
¤
k
)
k
.
O
§
E
Ñ
k
.
m
8
¶
•
|
^
ÿ
À
Ý
Ó
Ô
ØC
5
Ú
•
3
5
½
n
§
‰
Ñ
½
n
y
²
"
'
…
c
“
ê
Ä
½
n
§
ÿ
À
Ý
§
k
.
ProofoftheFundamentalTheoremof
AlgebraBasedonTopologicalDegree
XueleiWang
CollegeofInformationScienceandEngineering,ShandongAgriculturalUniversity,Tai’an
Shandong
Received:Nov.28
th
,2021;accepted:Jan.3
rd
,2022;published:Jan.10
th
,2022
Abstract
Inthepaper,proofoffundamentaltheoremofalgebraisobtainedbytopological
©
Ù
Ú
^
:
Æ
Z
.
Ä
u
ÿ
À
Ý
“
ê
Ä
½
n
y
²
[J].
n
Ø
ê
Æ
,2022,12(1):14-19.
DOI:10.12677/pm.2022.121002
Æ
Z
degreetheory.Fristweconstructahomotopyequation,whichcanreduceacomplex
problemintoasimplerone.Throughaprioriestimateforthepossiblesolutionsof
homotopyequation,wegainaboundedopenset;thenweprovethetheorembythe
homotopyinvarianceandexistencetheoremoftopologicaldegree.
Keywords
FundamentalTheoremofAlgebra,TopologicalDegree,PrioriBound
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.
Ú
ó
“
ê
Ä
½
n
´
p
“
ê
¥
š
~
-
‡
˜
‡
½
n
.
§
²
;
y
²
9
Ù
u
Ð
{
¤
§
Œ
±
ë
•
©
z
[1].
3
˜
„
p
“
ê
á
,
X
[2]
¥
Ñ
´
•
k
½
n
S
N
§
v
k
y
²
.
“
ê
Ä
½
n
•
•
{
ü
y
²
´
|
^
E
C
¼
ê
¥
Liouville
½
n
[3]
½
ö
Rouche
½
n
[3]
.
©
[4]
§
o
(
E
C
¼
ê
¥
Ú
Ä
–
“
ê
¥
|
^
³
Û
u
+
n
Ø
y
²
¶
©
[5][6]
|
^
)
Û
¼
ê
5
Ÿ
§
©
O
^
•
Œ
!
•
n
!
…
Ü
È
©
½
n
!
3
ê
½
n
‰
Ñ
“
ê
Ä
½
n
y
²
¶
©
[7]
^
“
ê
ÿ
À
¥
Ó
Ô
+
Ú
Ä
+
Ó
½
n
(
Ø
§
¿
r
½
n
(
Ø
í
2
•
˜
„
¼
ê
a
¶
©
[8]
¦
^
Ó
N
+
Ú
N
Ý
ó
ä
§
‰
Ñ
“
ê
Ä
½
n
ü
«
“
ê
ÿ
À
y
²
.
°
%
3
©
[9]
¥
§
|
^
à
g
‚
5
•
§
|
)
n
Ø
‰
Ñ
“
ê
Ä
½
n
˜
‡
{
'
“
ê
y
²
.
©
Ø
Ó
u
þ
ã
E
C
¼
ê
Ú
“
ê
ÿ
À
•{
,
·
‚
^
š
‚
5
•
¼
¥
ÿ
À
Ý
[10][11]
Ó
Ô
ØC
5
Ú
Œ
)
5
§
‰
Ñ
½
n
y
²
.
|
^
Ó
Ô
ØC
5
Œ
±
r
˜
‡E
,
¯
K
z
•
˜
‡
ƒ
é
{
ü
¯
K
.
A^
ÿ
À
Ý
'
…
´
k
.
m
8
E
§
•
Ò
´
Ó
Ô
•
§
˜
ƒ
Œ
U
)
k
.
O
.
2
|
^
{
ü
•
§
Ý
Ø
u
"
§
•
§
)
•
3
5
§
=
n
g
õ
‘
ª
3
E
ê
‰
Œ
S
–
k
˜
‡Š
.
3
©
Ù
•
§
‰
Ñ
A^
¢
~
.
“
ê
Ä
½
n
[1]
3
E
²
¡
þ
§
n
g
õ
‘
ª
f
(
z
) =
z
n
+
a
1
z
n
−
1
+
···
+
a
n
−
1
z
+
a
n
(1)
–
k
˜
‡Š
.
e
¡
·
‚
‰
Ñ
ÿ
À
Ý
˜
ý
•
£
.
DOI:10.12677/pm.2022.12100215
n
Ø
ê
Æ
Æ
Z
ÿ
À
ݽ
Â
[11]
Ω
´
R
m
¥
k
.
m
8
§
f
∈
C
2
(
¯
Ω
,
R
m
),
p
∈
R
m
,
p/
∈
f
(
∂
Ω).
p
´
f
K
Š
ž
,
½
Â
N
f
3
Ω
¥
'
u
p
:
ÿ
À
Ý
•
deg(
f,
Ω
,p
) =
k
X
j
=1
sgn
J
f
(
x
j
)
Ù
¥
J
f
(
x
)
L
«
N
f
Jacobi
1
ª
,
x
j
(
j
= 1
,
2
,
···
,k
)
L
«
•
§
f
(
x
) =
p
3
Ω
S
)
.
Kronecker
•
3
½
n
[11]
Ω
´
R
m
¥
k
.
m
8
,
f
:
¯
Ω
→
R
m
ë
Y
,
p /
∈
f
(
∂
Ω).
e
deg(
f,
Ω
,p
)
6
= 0,
K
f
(
x
) =
p
3
Ω
S
7
k
)
.
ÿ
À
Ý
Ó
Ô
ØC
5
[11]
H
:
¯
Ω
×
[0
,
1]
→
R
m
ë
Y
§
-
h
λ
(
x
) =
H
(
x,λ
).
e
p/
∈
h
λ
(
∂
Ω).
∀
0
≤
λ
≤
1,
K
deg(
h
λ
,
Ω
,p
)
ð
u
~
ê
(
é
u
∀
0
≤
λ
≤
1).
ÿ
À
ݽ
Â
!
5
Ÿ
9
5
Ÿ
y
²
§
Œ
±
ë
•
©
z
[11].
l
1912
c
Brouwer
M
á
ÿ
À
Ý
n
Ø
§
Ù
m
²
Leray
!
Schauder
ê
Æ
[
Ø
ä
í
2
õ
§
†–
y
3
§
ÿ
À
Ý
•{
®
¤
•
ï
Ä
š
‚
5
¯
K
Ä
•{
ƒ
˜
.
|
^
ÿ
À
Ý
n
Ø
)û
¯
K
§
'
…
3
uk
.
m
8
Ω
E
§
•
Ò
´
•
§
¤
k
Œ
U
)
k
.
O
.
e
©
¥
§
·
‚
r
deg(
f,
Ω
,p
)
{P
•
D
(
f,
Ω
,p
).
2.
“
ê
Ä
½
n
y
²
y
²
N
f
:
R
2
'
C
→
R
2
§
…
f
´
C
∞
a
¼
ê
.
-
g
:
R
2
'
C
→
R
2
,g
(
z
) =
z
n
−
1
.
•
Ä
Ó
Ô
•
§
h
λ
(
z
) =
H
(
z,λ
) =
λf
(
z
)+(1
−
λ
)
g
(
z
) =
z
n
+
λ
(
a
1
z
n
−
1
+
···
+
a
n
)+(1
−
λ
)
,λ
∈
[0
,
1](2)
K
H
:
R
2
×
[0
,
1]
→
R
2
´
C
∞
a
¼
ê
§
…
H
(
·
,
1) =
f
,
H
(
·
,
0) =
g
.
e
¡
E
k
.
m
8
Ω,
=
y
²
H
(
z,λ
) = 0
¤
k
Œ
U
)
´
k
.
.
5
¿
λ
∈
[0
,
1],
z
n
+
λ
(
a
1
z
n
−
1
+
···
+
a
n
)+(1
−
λ
) = 0
,
þ
ª
ü
>
§
|
z
|
n
≤|
a
1
|·|
z
|
n
−
1
+
···
+
|
a
n
|
+1(3)
e
Ó
Ô
•
§
)
Ã
.
§
é
?
¿
ê
m
§
Ñ
•
3
λ
m
∈
[0
,
1]
§
z
m
∈
C
§
¦
H
(
z
m
,λ
m
) = 0
…
|
z
m
|≥
m,
DOI:10.12677/pm.2022.12100216
n
Ø
ê
Æ
Æ
Z
ò
z
m
“
\
(3)
ª
§
…
ü
>
Ó
Ø
±
|
z
m
|
§
1
≤|
a
1
|·|
z
m
|
−
1
+
···
+
|
a
n
|·|
z
m
|
−
n
+
|
z
m
|
−
n
≤|
a
1
|·
m
−
1
+
···
+
|
a
n
|·
m
−
n
+
m
−
n
→
0(
m
→∞
)
g
ñ
.
Ï
d
Ó
Ô
•
§
H
(
z,λ
)=0
¤
k
Œ
U
)
´
k
.
,
=
•
3
M >
0,
¦
|
z
|
<M
,
-
r
= max
{
M,
2
}
,
B
(
r
)
L
«
±
‹
I
:
•
%
,
Œ
»
•
r
m
,
K
•
§
¤
k
)
z
∈
B
(
r
).
B
(
r
)
Œ
Š
Ω.
Ï
d
p
= 0
/
∈
h
λ
(
∂
Ω).
e
¡
O
Ž
{
ü
•
§
g
(
z
) =
z
n
−
1 = 0
3
z
∈
B
(
r
)
¥
ÿ
À
Ý
§
=
D
(
g,B
(
r
)
,
0).
g
(
z
) = 0
k
n
‡Š
§
=
1
n
g
•
Š
§
P
•
z
k
=
e
i
2
kπ
n
,
(
k
= 0
,
1
,
2
,
···
,n
−
1)
|
^
E
ê
†k
S
¢ê
éé
A
§
r
E
²
¡
þ
¼
ê
w
‰
R
2
→
R
2
N
,
=
z
=(
x,y
)
→
g
(
z
)=
(
u
(
x,y
)
,v
(
x,y
)),
O
Ž
Jacobi
1
ª
Î
Ò
.
J
g
(
z
) =
∂
(
u,v
)
∂
(
x,y
)
=
u
x
u
y
v
x
v
y
=
u
x
v
y
−
u
y
v
x
Ï
•
g
(
z
)=(
u
(
x,y
)
,v
(
x,y
))
´
)
Û
¼
ê
§
¤
±
u
(
x,y
)
,v
(
x,y
)
÷
v
…
Ü
-
i
ù
•
§
§
=
u
x
=
v
y
,
u
y
=
−
v
x
,
Ï
J
g
(
z
) =
u
2
x
+
v
2
x
=
|
g
0
(
z
)
|
2
=
|
nz
n
−
1
|
2
=
n
2
§
D
(
g,B
(
r
)
,
0) =
n
X
k
=1
sgn
J
g
(
z
k
) =
n
X
k
=1
sgn
n
2
=
n
Š
â
ÿ
À
Ý
Ó
Ô
ØC
5
§
Œ
D
(
f,B
(
r
)
,
0) =
D
(
H
(
·
,
1)
,B
(
r
)
,
0) =
D
(
H
(
·
,
0)
,B
(
r
)
,
0) =
D
(
g,B
(
r
)
,
0) =
n
6
= 0
d
ÿ
À
Ý
Kronecker
•
3
½
n
§
f
(
z
) = 0
–
k
˜
‡Š
.
5
µ
Ó
Ô
•
§
e
Š
z
n
= 0
§
K
0
´
•
§
n
-
Š
§
Ï
•
3
0
?
ê
Š
´
"
§
Ï
d
0
´
.
Š
§
Ø
U
†
O
Ž
ÿ
À
Ý
§
I
‡
^
K
Š
%
C
§
Ø
X
†
ü
Š
O
Ž
Ý
•
{
ü
.
3.
~
f
e
¡
Þ
~
`
²
(
Ø
(
5
.
f
(
z
)=
z
5
+
z
3
+6
z
3
B
(1) =
{
z
:
|
z
|
<
1
}|
S
w
,
k
Š
z
=0.
e
¡
^
ÿ
À
Ý
•{
5
y
²
ù
˜
(
Ø
.
E
Ó
Ô
•
§
h
λ
(
z
) =
H
(
z,λ
) =
λf
(
z
)+(1
−
λ
)
g
(
z
) =
λ
(
z
5
+
z
3
+6
z
)+(1
−
λ
)
·
6
z,λ
∈
[0
,
1]
DOI:10.12677/pm.2022.12100217
n
Ø
ê
Æ
Æ
Z
z
∈
B
(1) ,
=
|
z
|
= 1
ž
,
|
H
(
z,λ
)
|
=
|
λ
(
z
5
+
z
3
)+6
z
|≥|
6
z
|−|
λ
(
z
5
+
z
3
)
|
≥|
6
z
|−|
z
5
+
z
3
|≥|
6
z
|−
(
|
z
|
5
+
|
z
|
3
|
)
≥
6
−
2 = 4
>
0
.
Ï
d
p
= 0
/
∈
h
λ
(
∂
Ω),
g
(
z
) = 6
z
3
m
8
B
(1)
S
•
k
˜
‡Š
z
1
= 0,
u
(
x,y
) = 6
x,v
(
x,y
) = 6
y
,
O
Ž
z
1
= 0
?
Jacobi
1
ª
.
J
g
(
z
) =
∂
(
u,v
)
∂
(
x,y
)
=
u
x
u
y
v
x
v
y
=
60
06
= 36
¤
±
D
(
g,B
(1)
,
0) =sgn
J
g
(
z
1
) = 1.
A^
Ó
Ô
ØC
5
,
•
D
(
f,B
(1)
,
0) =
D
(
H
(
·
,
1)
,B
(1)
,
0) =
D
(
H
(
·
,
0)
,B
(1)
,
0) =
D
(
g,B
(1)
,
0) = 1
6
= 0
.
d
ÿ
À
Ý
Kronecker
•
3
½
n
§
f
(
z
) = 0
3
m
8
B
(1)
S
–
k
˜
‡Š
.
y
(
Ø
(
5
.
—
©
I
[
g
,
‰
Æ
Ä
7
(No.61573228)
!
ô
€
Ž
g
,
‰
Æ
Ä
7
(No.BK20181058)
|
±
.
ë
•
©
z
[1]
X
Z
.
“
ê
Ä
½
n
ï
Ä
{
¤
[D]:[
a
¬
Æ
Ø
©
].
Ü
S
:
Ü
Œ
Æ
,2021:53-61.
[2]
®
Œ
Æ
ê
Æ
X
A
Û
†
“
ê
ï
¿
c
“
ê
|
.
p
“
ê
[M].
®
:
p
˜
Ñ
‡
,2003:
27-28.
[3]
¨
Œ
.
E
C
¼
ê
Ø
[M].
®
:
p
˜
Ñ
‡
,2013:124-125,258-261.
[4]
Â
ï
.
“
ê
Æ
Ä
½
n
A
«y
²
•{
[J].
ê
ÆÆ
S
†
ï
Ä
,2019(10):84-85.
[5]
•
U
n
.
E
¼
ê
3
“
ê
Ä
½
n
y
²
¥
A^
[J].
É
²
“
‰
Æ
Æ
(
g
,
‰
Æ
‡
),2004,3(3):
277-278.
[6]
‰
ø
.
“
ê
Ä
½
n
E
©
Û
y
{
[J].
Œ
Æ
ê
Æ
,2005,21(4):111-114.
[7]
š
ý
ù
,
p
¬
V
.
“
ê
Ä
½
n
ÿ
À
y
²
9
í
2
[J].
³
e
ì
Œ
ÆÆ
(
g
,
‰
Æ
‡
),2018,
39(4):17-20.
[8]
Ü
œ
.
“
ê
Ä
½
n
ÿ
À
y
²
[J].
®
Ê
˜
Ê
U
Œ
ÆÆ
,1994,20(1):111-114.
[9]
°
%
.
õ
‘
ª
-
Š
O
ª
Ú
“
ê
Ä
½
n
{
y
[J].
ê
Æ
¢
‚
†
@
£
,2009,39(11):
128-132.
DOI:10.12677/pm.2022.12100218
n
Ø
ê
Æ
Æ
Z
[10]Mawhin,J.(1979)TopologicalDegreeMethodsinNonlinearBoundaryValueProblems.
ACBMSRegionalConferenceSeriesinMathematics
,
40
,42-50.
https://doi.org/10.1090/cbms/040
[11]
H
Œ
.
š
‚
5
•
¼
©
Û
[M].
L
H
:
ì
À
‰
Æ
E
â
Ñ
‡
,2001:87-135.
DOI:10.12677/pm.2022.12100219
n
Ø
ê
Æ