393Vol.39,No.319965
ACTA MATHEMATICA SINICA
May,1996
(
510400)
(
330047)
:
a ,
a
R
(K,P )
A
R A ={(ξα)|ξα∈
R,α∈A };
K [X α]{X α}α∈A
K
A ,
|A |
A
A ∞:=∪n ∈
A n ,
A n
A
n
A
|A ∞|=|A |.
K [X α]
J
n
f i ∈K [X α],p i ∈P ,n
i =1
p i f 2i ∈
J
f i ∈J ;
K [X α]
J
J ∩S =∅,S := 1+m i =1
p i f 2i m
f i ∈K [X α],p i ∈P
.
I
K [X α]
U
R A
V R (I )=
(ξα)∈R A |f ∈I,f (ξα)=0
;J R (U )={f ∈K [X α]|(ξα)∈U,f (ξα)=0};R
√I = f ∈K [X α]|m,f i ∈K [X α],p i ∈P
f
2m
+
p i f 2i ∈I
.
A
J R (V R (I ))=R √
I ,
什么是散文
D.W.
Dubois [2]
J.J.Risler [6]
1969
1970
A
S.Lang
[4]
Hilbert
1994
5
24
3
337
1
A
1(
)
(S 1)I
K [X α]
I
V R (I )=∅.
(S 2)I
J R (V R (I ))=R
√I .
(S 3)
M
K [X α]
M
(ξα)∈R A M =J R {(ξα)}.(S 4)
{θα}α∈A K D =K [θα]K
{θα}α∈A
M
D
D/M (K,P )
(S 5)
D
(S 4),
K
L
ψ
D
L
(S 1)⇒(S 2):J R (V R (I ))⊃R √
I
“⊂”.
f ∈J R (V R (I )),
t ,
K [X α][t ]
I =(I,1−tf ),
V R (I )=∅.S 1
f i ∈I ,h i ,h ,g
j ∈K [X α][t ]
p j ∈P f i h i +(1−tf )h =1+ p j g j 2
.t =1/f ,f
f 2n
f ∈R
√I .
(S 2)⇒(S 3):
1
B/C
I
B
I
B
I ∩C
C
2
R/K
{ξα}α∈A
R
R [ξα]/K [ξα]R
K [ξα]
R [ξα]
K [ξα]
r ∈R
(S 2)⇒(S 3)
(ξα)∈R A
M =J {(ξα)},
M
K [X α]
(X α−ξα)
{X α−ξα}α∈A R [X α](X α−ξα)∩K [X α]=
J R {(ξα)}.R [X α]/(X α−ξα)∼=R
(X α−ξα)R [X α]12J R {(ξα)}K [X α]K [X α]
M K [X α](1)=M =R √
M =J R (V R (M )),V R (M )=∅.
(ξα)∈V R (M ),
J R {(ξα)}⊃M .
J R {(ξα)}
M
M =J R {(ξα)},(ξα)∈R A .
(S 3)⇒(S 4):
I
K -
K [X α]→K [θα]
K [θα]I
K [X α]
M
M /I =M
K [X α]
M
K [X α]
(S 3)
(ξα)∈R A ,M =J R {(ξα)}.(S 2)⇒(S 3)
M
K [X α]
D/M ∼=K [X α]/M ∼=K [ξα]K ⊂D/M ⊂R ,
R/K D/M (K,P )
(S 4)⇒(S 5):I
K -K [X α]→K [θα]
K -
ρ:
K [X α]/I →K [θα].
I
K [X α]
Zorn
K [X α]
J ⊃I .
338
39
(S 4),E =K [X α]/J
(K,P )
P
P
E
E
(E,P )
E
(K,P )ψ(K )
L
ψ(K ),
ψψ(K )(ψ(K ),ψ(P ))
[5]
(3.9)
Zorn
ψ
E
ψ,
D ρ−1
−→K [X α]/I τ−→K [X α]/J =E ψ|E
−→L,
τ
f +I →f +J .σ=ψ|E ·τ·ρ−1,σ
D
L
σ|K =ψ,
σ
(S 5)⇒(S 1):
I
I ∩S =∅,
V R (I )=∅.
I
K [X α]
I ∩S =∅.
Zorn
{J |J
K [X α]
J ∩S =∅,J ⊃I }
Q
Q
Q
n
p i ∈P ,f i ∈K [X α]\Q ,f =
n i =1
p i f 2i ∈Q.
(1)
Q +f 1·K [X α] Q .
Q
(Q +f 1·K [X α])∩S =∅.
m
g ∈Q ,h,h i ∈K [X α],q i ∈P ,g +hp 1f 1=1+m
i =1q i h 2i .
(1)
g (g +2hp 1f 1)=1+ r j g 2
j +h 2p 1 −f +n i =2
p i f 2i ,r j ∈P ,g j ∈K [X α].
g (g +2hp 1f 1)+h 2
p 1f =1+
r j g 2
j
+
n
i =2
(p 1p i )(hf i )2∈Q ∩S ,Q
Q
K [X α]/Q ≡K [ξα]
ξα:=X α+Q .
(S 5)
K [ξα]
(K,P )
R
K -
ψ.
(ψ(ξα))∈R A
Q
I
V R (I )=∅.(K,P )|A |
1
(
)
(K,P )
|A |
3D,D
D /D
D
D
2
(K,P )
|A |
(i)A (ii)A
tr.d K =b ,
b >a ,a
A
tr.d K
K
Q
(i)
(ii)
A
b ≤a .(K,P )
R c ≤a (b
).
C
A
c
C
R R ={ξβ|β∈C }.pgl2008
Y
R
R (Y )
Ω={Y }∪{1/(Y −ξβ)|β∈C }∪ 1 (Y −ξβ)2+ξ2γ β,γ∈C .
3
339
Ωc ,C
Ω={θβ|β∈C }.
D =R [Ω],
R [Y ]⊂D ⊂R (Y ).
f ∈˙D
,
R
R [Y ]
(
Y −ξβ)
(
(Y −ξβ)2+ξ2
γ)
f −1∈D ,
D =R (Y )
2
D
K [Ω]
3
K [Ω]
R
(K,P )R (Y )
R
R (Y )(K,P )
K ⊂K [Ω]⊂R (Y ),
K [Ω]
(K,P )
(0)
K [Ω]
K [Ω]/(0)∼=K [Ω]
(K,P )
Y ∈K [Ω](=K [θβ|β∈C ])
K
(S 4)
2
(ii)
[4]
K =R =R (),A =N (
),
tr.d R >|N |.
tr.d R ≤|N |,
R
Q T ,
|T |≤|N |,
T
Q (T )
R
Q (T )
R [X 1,···,X j ,···]
I ,
(X 1−j )X 2
j −1,
j =2,3,···.
I
1+
m i =1
f 2i ∈I ,f i ∈R [X 1,···,X j ,···],
M ,
1+
m i =1
f 2
i =M j =2
g j (X 1−j )X 2
j −1 ,
(2)
f i ,
g j ∈R [X 1,···,X M ](
g j
).
α1=M +1,αj =
j =2,3,···,M .
(2)
X j =αj (j =1,2,···),
1+m i =1
f 2i (α1,···,αM )=0,
I
V R (I )=∅.
(a 1,···,a j ,···)∈R
I
(a 1−j )a 2j −1=
0,j =2,3,···,
(a 1−j )a 2j =1>0,
a 1−j >0,a 1>j ,j =2,3,···.
R
1
(S 1)
(K,P )
|A |
f ∈K [X α],D R (f )=
{(θα)∈R A |f (θα)=0};V R (f )= (ξα)∈R A f (ξα)=0
.
D R (f )∩D R (g )=D R (fg ),
R A
Zariski
{D R (f )|f ∈K [X α]}
Zariski
R A
V R (f ),f ∈K [X α],
(K,P )|A |
3
(K,P )
|A |
R A
Zariski
{F λ|λ∈Λ}
R A
∩λ∈A
F λ=∅.
R A
F λ=V R (f λ),f λ∈K [X α],λ∈Λ.
J
{f λ|λ∈Λ}
K [X α]
V R (J )=∩λ∈Λ
V R (f λ)=∅.
1
(S 1)
J
K [X α]
m,n
p i ∈P ,h i ,g j ∈K [X α]
340
39
f λj
1+
m i =1
p i h 2i
=
n j =1
f λj
g j .
(3)
R
(3)
∅=V R (f λ1,···,f λn )=n
∩j =1
V R (f λj )=n
∩j =1
F λj ,
R A
1
(S 1)
I
K [X α]
V R (I )=∅.
R A
{V R (f )|f ∈I },
∩f ∈I
护理学杂志V R (f )=V R (I )=∅.
R A
f 1,···,f s ∈I
∅=s
∩i =1
V R (f i )=V R (f 1,···,f s ).
(4)M
f i ∈K [X α1,···,X αM ],i =1,···,s .I
f 1,···,f s
K [X α1,···,X αM ]
(4)
M
R M
f 1,···,f s
I
I
K [X α1,···,X αM ]
g i ∈K [X α1,···,X αM ],p i ∈P
m ,
1+m i =1
p i g 2i ∈I ⊂I .
I
K [X α]
(S 1)
2
R A
Zariski
1
R
u,v ,
u <v ,
]u,v [,]−∞,v [
]u,+∞[
{x ∈R |f i (x )>0,i =1,···,m ;g (x )=0},
(∗)
f i ,
g ∈K [X ],i =1,···,m .
]u,+∞[]−∞,v [
ϕ(X )u K R
ϕ(X )=(X −u 1)···(X −u s ) (X −v 1)2+w 21
··· (X −v t )2+w 2t ,u i ,v j ,w j ∈R ,
w j =0,i =1,···,s ;j =1,···,t .
u 1<u 2<···<u s ,u =u r ,1≤r ≤s .
ψ(Y,X ):=Y 2(s +2t )·ϕ(X −1/Y 2)
=
s i =1
(X −u i )Y 2
−1 ·t
j =1
XY 2−v j Y 2−1 2
+w 2j Y 4
∈K [Y,X ].
3341 x∈]u,+∞[,ψ(Y,x)2r x K
∃Y1,···,Y2r(ψ(Y1,x)=0∧···∧ψ(Y2r,x)=0∧Y1<···<Y2r).
([5] 5.1),(∗)
ψ(Y,x)=0R2r x
2R]u,v[,u=−∞,v=+∞,m,
]u,v[R1+m K-R
1,]u,v[(∗)
1>0,m.(∗)
R1+m R
K-V:f i(X)Y2i−1=0,i=1,···,m;g(X)=0.
R1+m R(x,y1,···,y m)→x V(∗).
3A{Sα}α∈A(
),α∈A,Sα=nα∪
kα=1Sα,k
α
,
nα∈N,α∈A,rα,1≤γα≤nα,
{Sα,r
α
}α∈A
Γ(A ,r)
A A r A Nα∈A ,1≤rα≤nα,
{Sα,r
α
}α∈A ,∪{Sα}α∈A\A
β∈A.kβ=1,···,nβSβ,k
β
{Sα}α=β
()Sβb,1≤b≤nβ, {Sβ,b}∪{Sα}α=βA0={β},rβ=b.(A0,r)∈Γ,ΓΓ“ ”(A1,r1),(A2,r2)∈Γ,(A1,r1) (A2,r2)
A1⊂A2,r2A1r2|A
1
=r1.
Γ ZornΓ(A∗,r).
A∗=A.β∈A\A∗,{Sα,r
α
}α∈A∗∪{Sα}α∈A\A∗
b∈N,1≤b≤nβ,{Sα,r
α
}α∈A∗∪{Sβ,b}∪{Sα}α∈A\A
A ={β}∪A∗.r A N r β=b,r α=rα,α∈A∗.
(A ,r )∈Γ,(A∗,r)A∗=A,
41,
(i)(K,P)|A|
(ii)R A
(iii)R A
A A
(i)⇒(ii).{Tα}α∈A R A2
α∈A,mα∈N,TαR1+mαK-VαR
VαK[X,Xα,1,···,Xα,m
多边主义
α
]Iα
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