(M, +) M
+ M
a,b,c
(M, +) a + (b + c) = (a + b) + c id+ M M
M
m + id+ = m
m y
id+ + m = m
∈
M M
a + b = b + a
(M, +) u y + u = id + u + y = id + U (M ) = M (M, +) a, b M
M U (M ) (M, +) (M, +) x
∈ M
+
−x ·
(M, +, ) M
• (M, +) • (M, ·) •
a,b,c
∈ M a · (b + c) = a · b + a · c
+
(a + b) c = a c + b c
·
·
(M, ) a b = id +
·
(M, ) a = id +
⇒
·
b = id +
·
(M
\ {0}, ·)
·
·
·
+
0 1 k
∈ N
Nk := 1, 2, . . . , k
{
M a, b M a b M
∈
S M ab M
− ∈
S
⊂
∈
} S =
∅
A
n
K
(K[x1 , . . . , xn], +, ) K
n
·
n
A
n A[n] xαi =
(αi1 , . . . , αin)
n αij j=1 x j
αi j
∈ N ∪ {0}
f A[x1 , . . . , xn ] ai A αi ai x ai = 0 f M(f ) r m M(f ) deg(m) = k=0 αk deg(f ) := max deg(m) m M(f ) A[x1 , . . . , xn]
∈ · ∈
f f
r i=0
f
∈
{
| ∈
A ai xαi
m deg(f ) = 1
}
A[x1 , . . . , xn ] r N αi =
∈
xαi f
∈ A[x , . . . , x ] 1
n
f
α A[x1 , . . . , xn ]
∩A
f A [n] deg(f ) < 1
f A
∈
max n,m i=0
{
} c xi i
·
·
(A, +, ) ci = ai + bi nm
f
∈
(A[x], +, ) nm f g =
·
(A[x1 , . . . , xn ], +, ) A[x1, . . . , xn] = A[x1 , . . . , xn−1 ][xn ]
·
bm+1 = n+m i i=0 ci x
n
bm+2
f + g = = = bn = 0 ci = j+k=i a j bk
···
∈ N n ≥ 2 n = 1
xn
A[x1 , . . . , xn−1 ] A[x1 , . . . , xn−1 ]
A[x1, . . . , xn] = A[x1 , . . . , xn−1 ][xn ]
f A[x1 , . . . , xn ]
∈
m
f =
ai xα1 i . . . xαnin , ai 1
i=1
f =
m αin i=1 bi xn
i
∈N
∈
bi = ai xα1 i
j Nn A[x1 , . . . , xn−1 ][xn]
∈
m
αij
αi n −1 n 1
1
A[x1 , . . . , xn ] = A[x1 , . . . , xn−1 ][xn ]
r2 i i=0 bi x1
r1 , r2 N ai , bi ar br = 0
= 0
1
n 1
A[x1 , . . . , xn]
·
(A, +, ) n = 1 0 g =
∈ N ∪ {0}, ...x − ∈ A[x , . . . , x − ]
∈ A
∈
1
2
∈ A
n
≥ 2
ar , br = 0 1
2
f =
f g = 0
·
A[x1 , . . . , xn] f / A A[x1 ]
f
∈
⊂ A[x , . . . , x ] deg(f ) ≥ 1 1
n = 1
∈
deg(1) < 1 A f A[x1 , . . . , xn ] A A[x1 , . . . , xn ] A[x1, . . . , xn−1 ] A[x1 , . . . , xn−1 ] A[x1 , . . . , xn ]
∈
\
\
ψ (a +1 b) = ψ(a) + 2 ψ(b) ψ(a) = b a A ψ ψ
∈ }
B
B
B
e
∈ Ker(ψ)
A
deg(1) A
·
∈
→ B
A
∈
1
·
→ B { ∈ B |
·
}
Im(ψ )
Ker (ψ)
Ker(ψ)
A
ψ (a) = c ψ(b) = d ∈ cd = ψ(a)ψ(b) = ψ(ab) c − d = ψ(a) − ψ(b) = ψ(a − b), a − b ∈ A ψ(ab) = cd ∈ I m(ψ) ψ(a − b) = a − b ∈ I m(ψ) Im(ψ ) e, f ∈ K er(ψ) ψ (e) = ψ(f ) = 0 ψ(e − f ) = ψ(e) − ψ(f ) = 0 ψ(ef ) = ψ(e)ψ(f ) = 0,
− f,ef ∈ K er(ψ)
ab
·
ψ:A
a, b A
∈
∈ A
f A[x1 ]
f g = 1 deg(f ) 1 deg(g) 0 A[x1 , . . . , xn−1 ]
{ ∈ |
c, d I m(ψ)
ab
Ker (ψ) a
(A, +1 , 1 ) (B, +2 , 2) ψ:A a, b A ψ(a 1 b) = ψ(a) 2 ψ(b) ψ(1A) = 1B Im(ψ ) = b ψ Ker(ψ) = a A ψ(a) = 0
·
A
ai xi1 =
A
n
g A [x1 ] deg(1) = deg(f ) + deg(g) 1
∈
r1 i=0
A[x1 , . . . , xn ]
·
(A, +, ) A A
∈A
b Ker(ψ) Ker (ψ)
∈
A ψ(ab) = ψ(a)ψ(b) = ψ(a)0 = 0 Ker(ψ)
A a A
A A/I = a + I a
I
{
∈
R
I A
| ∈ A}
I
⊂ R
a R
∈
ψ ψ(a
| ∈ I }
{
→ R/I
B
π(a) = a + I I ψ:A
→ B
Im(ψ )
ψ : A/Ker(ψ) I m(ψ) ψ(x + Ker(ψ)) = ψ(x) x A a + Ker(ψ ) = b + Ker(ψ), a , b A a b Ker(ψ) b) = ψ(a) ψ(b) = 0 ψ (a) = ψ(b) ψ(a + Ker(ψ)) = ψ(b + Ker(ψ)) b I m(ψ) a A a A ψ (a) = b b = ψ(a + Ker(ψ)) ψ (a + Ker(ψ)) = ψ(b + K er(ψ)) ψ(a) = ψ(b) a b Ker(ψ) a + Ker(ψ) = b + Ker(ψ)
→
−
a + I = a + i i
π:R
A A/Ker(ψ)
⊂
∀ ∈ − ∈
∈
−
∈
∈
∈
− ∈
K[n]
M =
∅ a
K
a, ∀a ∈ M a, b ∈ M a,b,c ∈ M a
b
a
b b a a b b c a ≺ b
a = b a a
a
c b b
≺ a b a,b, ∈ M
a = b
M
K[x1 , . . . , xn ] M
a, b
∈M
a
b
a c
· b · c, ∀c ∈ M
M
a, b M
∈
M
m, n
n n = j=0 x j βj k N γ p = 0
∈
m
LR
M
m Ln m p < k γ k > 0
n
n
k
∈ N
γ p = 0
∈ M
m = γ = β
−α
k < p n γ k > 0
f
∈ K[x , . . . , x ] f ml(f ) := max {m|m ∈ M(f )} f tl(f ) := ml(f ) · cl(f ) ml(f )
f cl(f )
p, q p b1
n αi i=0 xi
p q
|
bn
x1 . . . xn t = x 1 c . . . xn cn
−
1
∈
f = q g + r
·
ml(g) ml(r)
q =
1
∈
n [n]
q
p = x1 a . . . x n a n t M
∈N
∈ K
q
·
n
q
f, g
n
p q = p t
n
1
ai , b i 0 i Nn p ci = b i ai 0 i
1
∈ K[x , . . . , x ] t ∈ K[x , . . . , x ] p, q ∈ M
1
g=0
[n]
q, r
∈ K
ml(f ) q = 0 ml(g ) ml(f ) tl(f 1 ) f i+1 = f i q i g f i = 0 f m
ml(g)
r = f f f 1 = f q 1 = tl(f 1 )/tl(g) f 2 ml(f 2 ) q i = tl(f i )/tl(g)
r = q = 0 f 2 = f 1
− q · g (f ) ml(g ) | ml(f )
ml(f )
f m = f m−1 q m−1 g f m−1
−
·
1
n
q i
− q − · g = f − − q − · g − q − · g m 1
m 2
f m−2 , f m−3 , . . . , f2 + q m−1 ) r = f m q = (q 1 + q 2 +
m 2
m 1
f = f 1 = (q 1 +q 2 + f = q g + r
· ··
·
q 1 , q 2 , r1, r2 ml(g) ml(r1 ) ml(g) ml(r2 ) m M(r1 ) r2 r2 = r1 M(r2 r1 ) M(r1 ) M(r2 ) r2 = r 1
·
− ·
i
ml(g) ml(f m )
f m = f m−1
f = q 2 g + r2 ml(g ) m (q 1 q 2 ) g = r2
1
| − ·
∈ − ⊆
− r
1
∪
··· +q − ) · g +f m 1
[n]
∈K
f = q 1 g + r 1 ml(g) ml(r1 ) q 1 g + r1 = q 2 g + r2 r1 ) tl(g ) tl(r2 q 2 = q 1
·
·
· −
|
r rg (f ) a 1 , a 2 , . . . , ak i
∈N
k
r
∈ K
[n]
q
f f = a1
·
f 1 , . . . , fk f 1 + ... + a k f k + r
·
m
ml(r )
g
f
[n]
g
[n]
∈ K
f
ml(f i )
∈ K
[n]
f
f = f 1 a1 + r1 r1 = f 2 a 2 + r 2 ri m>0 f = a 1 f 1 + a2 f 2 +
· · ·
·
∈ K tl(f ) tl(r ) f = f · a + f · a 1
···
∈ K ∈ K
1
1
1
2
+ r 2
2
[n]
a1 , r 1 a2 , r2
f i tl(f k ) tl(rm ) + ak f k + rm
k
·
∈ N
[n]
f
n
r f
G = f 1 , . . . , fk
{
y
≺ x
rG (f )
}
f = xy 2 x, g1 = xy f g1 f = yg 1 + g 2 f = xg 2 + 0.g1 + x2 x
g2 = y 2
−y
−
g2
− x ∈ K[x, y] g2 2
x
−
g1
− x = 0
K[x1 , . . . , xn]
I A I
·
A
⊂
(A, +, ) 0 I
∈ a, b ∈ I a ∈ I b ∈ A
a + b I
∈
a b I
· ∈
A
A
{0}
a1 , . . . , as
A
⊂ A A
a1 , . . . , as a 1 , a 2 , . . . , ak
∈ A
a , a , . . . , a 1
2
k
{a · b + a · b + · ·· + a · b | b , b , . . . , b ∈ A}, a ,a ,...,a a , a , . . . , a 1
1
2
2
1
k
2
k
1
2
k
k
1
2
k
a 1 , a 2 , . . . , ak
f 1 , . . . , fk
A ∈ A f , . . . , f 0 = a · 0 + · ·· + f · 0 ∈ f , . . . , f 1
k
1
a, b
∈ f , . . . , f 1
k
k
a1 , . . . , ak , b1, . . . , bk a = f 1 a1 +
·
··· + f · a k
k
1
k
0
∈ A
b = f 1 b1 +
·
··· + f · b .
a + b = f 1 (a1 + b1 ) + a+b
· c ∈ A
∈ f , . . . , f 1
k
k
· ·· + f · (a + b ), k
a c = f 1 a1 c +
·
ai c A
i
· ∈
I
∈N
k
J I + J
I + J = a + b a
{
| ∈
1
I, J I , b J
a1 , b1
∈
A
∈ I
k
∈
a2 , b2
∈ J
k
· ∈
k
k
1
I J
A
0 I 0 J a, b I a, b J I J I a c J a c I J
0
∈ I ∩ J a + b ∈ I ∩ J I ∩ J a, b ∈ I + J
∈ ∈ ∈ ∈ · ∈ · ∈ ∩
a = a 1 + a2
A
k
I + J
∩
∈ } ∈
k
· · · ·· + f · a · c a · c ∈ f , . . . , f f , . . . , f I ∩ J
I, J a, b I J a + b I a + b J c A a c
∈ ∩
k
b = b 1 + b2
a + b = (a1 + a2 ) + (b1 + b2 ) = (a1 + b1 ) + (a2 + b2 ) I J a I + J b A
∈
a1 + b1
∈ I
∈
a2 + b2 J a b I + J
· ∈
a + b I + J
∈
∈
S I S
S = I
⊂
I I K[x1, . . . , xn ]
I
I
A
L
⊂ I L ⊂ I L ⊂ M ⊂ A L ⊂ M a ∈ L I L ⊂ I a ∈ I p ∈ L p M p ∈ M L ⊂ M
a
L
L ⊂ I L
I
A
I
S
S
I
I
{g , . . . , g } 1
G = gi
s
gi gi I
X S
⊂
∈ I, ∀i ∈ N
S
s
S X =
s i=1 S i
∪
I
S i
⊂ S
G
K[x]
I = K[x] deg(g) g g = f q g + rg
{
| ∈ I }
·
I = 0
f
∈ I
deg(f ) g q g , rg K[x] deg(f ) > deg(rg ) rg I f, q g g I = 0 I = K[x] rg = 0 g f
tl(rg ) deg(f )
tl(g)
{}
I
∈
∈
∈
{ }
· ∈ I ∈
K[n]
I
⊂ K
f 1
∈ I
[n]
I = 0
{}
I = 0
{ }
K[x]
[n 1]
K − f 1 = g1 xan xn
·
f 1
g1
[n 1]
−
∈ N I = f f ∈ I \ f I = f , f f = g x ∈ I \ f , f . . . , f −
1
∈ K
a1
1
2
f 2 = g 2 xan2
·
r
∈ N
u N G = g1 , g2 , . . .
{
∈
a2
∈ N
I = f 1 , . . . , fu
}
1
1
ar r n
r
1
2
2
r 1
f r
M = m1 , . . . , ms
{
G
J = G K[n−1] J = M
} ⊂ J
⊂
M
M G gi = p i m j gi = p i q j gk K M
j i = k
i = k M G
· ·
∈ N k ∈ N q ∈ K
M j N
i
·
∈
g2 ∈ K −
ar
G
pi
[n 1]
j
∈
[n 1]
−
pi K [n−1] m j = q j gk
∈
·
pi q j = 1 G
·
M
f G
i>k
f i = g i xani = p i q j gk xani = p i q j gk xani −ak xank = p i q j xani −ak f k .
·
· · ·
· · ·
f i f 1 , . . . , fk , . . . , fi −1
·
f k
i>k
· ·
·
f i i
[n]
K
f M(f )
⊂ I
∈K
[n]
I = m1 , . . . , ms
f I
[n]
q 1 , . . . , qs K ml(f ) = ml(m1 f 1 = f tl(f ) f 1 I f 1 f I M(f ) I
∈
∈
−
∈ ⊂
·
f I f = m 1 q 1 + + ms q s q 1 + + ms q s ) = m1 q 1
· ··
·
·
·· ·
· ·
f
∈ I
ml(f )
f
ml(I )
I in(I ) =
ml(I )
I
K
in(I )
K[n]
G u
v
∈ ml(G)
{ml(f ) | f ∈ I }
ml(I )
[n]
u
=
I
p
G
I
in(I ) = f I
∈ f ∈ I
K[n]
G =
= max 1is
{ml(q ) ·
I
⊆
ml(G) rG (f ) = 0 f =
}
g ml(I ) m1 , . . . , ms ml(I ) f i I hi K [n] f i I u G ml(f i ) g ml(G) ml(I ) ml(G) ml(I )
∈ ∈ ∈
∈ ml(I )
ml(gi )
1
v
{g , . . . , g } ⊂ I 1
I
⇒ 2
⊂
s i=1 q i
∈ m g = ∈ m = h · ml(f ) ∈ ∈ ml(u) ⊂ ml(G) s i=1
i
i
i
·g
i
ml(f )
i
i
∈N G f ∈ ml(G) G ⊂ I
i
s
I
i
ml(G)
⊂
ml(I )
q 1, q 2 , . . . qs , rG(f ) K[n] i Ns ml(gi ) G I q I ml(rG (f )) ml(I ) ml(I ) = ml(G) rG (f ) = 0 G I
f I s f = i=1 gi q i + r G (f ) ml(rG(f )) q = si=1 gi q i q f I rG (f ) = f q I gi ml(rG (f )) rG(f ) = 0 f 2
⇒ 3
∈ ·
∈
·
− ∈
r (f ) = 0 ⇒ 4 f ∈ I q , q . . . , q ∈ K {ml(q ) · ml(g )} 4 ⇒ 1 f ∈ I ml(f ) = max 3
1
max1is
2
i
f
G [n]
s
∈
f =
i
{
s i=1 q i
1is ml(q i )
ml(G)
G
I
· g
⊂
⊂
ml(f ) = ml(
i
G
s i=1 q i
· g ) = i
· ml(g )} i
K[n] I K[n] in(I ) = in( G ) = ml(G)
⊂
I
∈ ∈ ∈
G = g1, . . . , gs
{
G
}
I
in(I ) =
ml(I ) = ml(G) = in(G) ⊂ in(G), J = G J ⊂ I G ⊂ I in(G) = in(I ) ml(G) ⊂ ml(G) ml(G) ⊂ ml(G) = in(G). ml(G)
⊂ ml(G)
ml(G)
in(J )
⊂ in(I )
G
⊂ G
f in( G ) f f ml(G) ml(G)
∈
f
∈
ml(I )
f
G ⊂ I
ml(G)
ml(G) K[n]
I, J
⊆ K
I = J I = J
I J = f I J f I ml(f ) ml(I ) ml(f ) in (I ) = in(J ) g) tl(f ) f tl(g) = tl(f ) tl(f (f g) + g = f J
∈
∈
\ ∅ ∈ \ ∈ − ≺
I I
ml(G) = in(J )
ml(I )
in( G ) =
in(I ) = in(J )
−
⊂ ml(I )
ml(I )
G
∈
ml(J )
[n]
⊂K
G I I = J
[n]
J I
⊂
I J
\
tl(f )
f J in(I ) = in(J ) I J f g I = J
∈
g
− ∈ J
\
G
I
J = G
⊂
G K[n]
⊂ I
∈ J g ∈ J
G
in(I ) = I
K[n]
I f I
∈
⊂K
[n]
∈ ml(I ) ⊂ G = {f , . . . , f } ml(f ) 1
s
J = in(I ) m1 , . . . , ms J J = m1 , . . . , ms J mi ml(f ) f i ml(f i ) = m i , i N s
∈
G
∈
I
K[n]
G
I
M
m1 m2 i
∈N
m1 = mmc(m1, m2 ) = ni=1 xci i
n
n i=1
M f g
f
g
tl(f )
tl(g)
· − mmc(ml(f ), ml(g)) · K[x, y]
xy
−y
g2 = y
2
f, g
S (f, g)
S (f, g) := mmc(ml(f ), ml(g))
−x
xy y S (g1 , g2 ) = (xy ) xy 2
2
y
xai i , m2 = ni=1 xbi i ci = max ai , bi
∈ K
≺ x
g1 =
2
2
2
K[x1 , . . . , xn ] f, g
∈ K[x , . . . , x ] S (f, g) = −S (g, f ) m , m ∈ M 1
1
n
2
S (m1 , m2) = 0
f ml(mmc(ml(f ), ml(g)) tl(f ) )
ml(S (f, g))
[n]
− − (xy ) y − x = y(xy − y) − x(y − x) = x − y . y 2
= mmc(ml(f ), ml(g))
≺ mmc(ml(f ), ml(g))
{
∈ }
2
f S (f, g) = mmc(ml(f ), ml(g)) ( tl(f )
·
g
−
m S (m1 , m2 ) = mmc(ml(f ), ml(g)) ( tl(m
·
q =
mmc(ml(f ),ml(g)) tl(f )
ml(f )
)=
f
tl(f )
m2
1 1
g
tl(g)
−mmc(ml(f ), ml(g)) · ( − ) = −S (g, f ) − ) = mmc(ml(f ), ml(g)) · ( − ) = 0
tl(g)
)
=
m1 m1
tl(m2 )
p ai i=1 xi
ml(g)
p i=1
=
p
mmc(ml(f ), ml(g)) =
m2 m2
xi bi
xi max{ai ,bi } .
i=1
· f ) = q · ml(f ) =
p γ i i=1 xi
ml(q
mmc(ml(f ), ml(g))
− a ) + a i
= ci
i
· f ) =
ml(q
f g = ml(mmc(ml(f ), ml(g)) tl(f ) mmc(ml(f ), ml(g)) tl(g) ) max ml(f ), ml(g) mmc(ml(f ), ml(g))
ml(S (f, g))
≺
γ i = (ci
{
−
}
≺
S (f, g) ml(S (f,
[n]
K rG(S (gi , g j )) = 0
g))
ml(f )
G = g1 , . . . , gs gi , g j G
{
∈
I rG (S (g1 , g2 )) = 0
f I G h1 , h2, . . . , hs K[n]
∈ N ) s
L =
gl
ml(f )
L=
H = i
ml(f )
f = h 1
∈ | ml(f ) k ∈ N
·
g g1 + ... + hs gs
g1 , g2
∈
ml(g)
·
f
≺ L
· g ) = ml(f ) L
ml(hk
k
·
f
L f L = ml(f )
H h2 g2
·
h1 g1 + h2 g2 = tl(h1 ) g1 + tl (h2 ) g2 + (h1
·
·
·
·
− tl(h )) · g + (h − tl(h )) · g 1
1
2
2
· g ) = tl(tl(h ) · g ) ml((h − tl(h )) · g ) = ml(h · g − tl(h ) · g ) ≺ L, 1
1
1
1
tl(h1 )
1
1
1
1
1
1
)·g tl(h ) · g + ). · g + tl(h ) · g = tl(h · g ) · ( tltl(h (h · g ) tl(h · g ) tl(h · g ) = −tl(h · g ) 1
2
2
1
1
1
1
1
1
1
1
2
2 2
2
· |
H
H f h1 g1
| ml(f )
L = h l gl = max (ml(hi gi ) L
i
H
tl(h1
∈ G
·
ml(f )
s
i
I
G
∈
{ | ml(h · g ) = L}
I
} ∈
G S (g1, g2) I
G
i
ml(g)
2
2
2
tl(h1 )
· g + tl(h ) · g = tl(h · g ) · ( 1
2
2
g = tl(h1 g1 ) ( tl(g + )
· ·
1
1
1
g2 tl(g2 )
)=
1
tl(h1
2
1
·
· S (g , g ). 2
s
pi gi + (h1
·
i=1
− tl(h )) · g + (h − tl(h )) · g + 1
ml((h2
1
1
2
2
2
hi gi .
·
i=3
− tl(h )) · g ) ≺ L 2
2
(h g ) ) ≺ L · mmc(ml ml(g ), ml(g )) 1 1
1
2
L
h j g j
f
·
≺ L
ml(h j
·g ) = L j
H
ml(f )
f ml(f )
G
1
f
ml(S (g1 , g2 )
H
2
S (g1 , g2 )
− tl(h )) · g ) ≺ L 1
2
2
s
·g )
mmc(ml(g1), ml(g2 ))
ml((h1
1
1
∈
G f =
·g tl(h )·g ·g ) + tl(h ·g ) ) =
tl(h1
tl(h1 g1 ) mmc(ml(g1 ),ml(g2 ))
rG(S (g1, g2 )) = 0 p1, . . . , ps K[n]
G
tl(h1 )
H
= L
I
G
G
G
G K[n]
I G
{
|
S = I
⊂
S
P = (g, g ) g, g
G = S
S I
∈ G, g = g }
P =
∅ (g, g ) P
P
∈
r = r G (S (g, g )) r=0
G = G
∪ {r }
P = P
∪ {(g, r) | g ∈ G}
G P = (g, g ) ∈ G
∅
G
G
K[n] I i
N
m
∈N
I m = I m+ j
[n]
⊂ K j ∈ N
,i
∈ N
I i
⊆ I , ∀i ∈ i+1
I = i∈N I i i j N I i f + g I I j I i f I i
∪
f + g f + g
∈ ⇒ ∈ ⊆
0 i = j
1
i
i
j
r
1
m
r
r
∈ I
⊆
[n]
∈ K f , . . . , f ∈ I f ∈ I , . . . , f ∈ I
I j
{ | ∈ N } {f , . . . , f } ⊂ I f , . . . , f = I ⊆ I ⊆ I , j I ⊆ I ⊆ I ⊆ I 1
I i
f, g I i I j
a
1
f k
m = max jk k
∈ I
∈
∈
∈ ⊆
r
0
i =j i > j f I j f I af I i I I i
∈ ∈
f , . . . , f = I
∈ I ⊆ I f ∈ I g ∈ I
1
m
r
j1
r
jr
m
m+ j
I m = I m+ j
j
∈ N
G G
i
∈
G1 = G N
Gi+1 = Gi
I j = I j+d
r1 , r2 , . . .
∪ {r } i ∈ N
I i = Gi
I ⊂ I d ∈ N
i
G j = G j+d G j
G j
i
P
− }
− x ∈ I
≺ x K[x, y] I = S
rS (f )
xy y S (g1 , g2 ) = (xy ) xy
2
S = g1 = xy y, g2 =
{
S
2
2
2
2
2
rS (S (g1 , g2 )) = x2 + x = 0
{
−
− − (xy ) y − x = y(xy − y) − x(y − x) = x − y , y
2
G = g1 , g2 , g3
i+1
∅
y y x f = xy 2 I
⊂ G ∈N
I P =
2
Gi j
i+1
g3 = x2 + x
}
G
r (S (g , g )) = 0 • S (g , g ) = g • S (g , g ) = (x y) − −(x y) − = x(xy −y)−y(x −x) = −xy+yx = 0 r (S (g , g )) = 0 • S (g , g ) = (x y ) − − (x y ) − = y (x − x) − x (y − x) = x − xy = xg − xg 1
2
1
3
2
3
3
G
2
xy y xy
1
2
x2 x x2
2
2
2 2 y2 x y2
2 2 x2 x x2
2
2
G
2
2
3
2
1
3
3
2
rG (S (g2, g3)) = 0
G
I
G
ml(g)
| ml(h)
I H = G
[n]
⊂ K \ {h}
G
⊂ I
I I
g, h
∈ G
f p ml(g)
g
g, h G p H ml( p) ml(f ) ml( p) = ml(h) ml(f )
∈ G
| ml(h)
∈ H
∈ I
∈
|
∈
|
H
p=h
I
[n]
I I
G
| ml(h) ml(g) | ml(f )
ml(g)
⊂ K
G
G
⊂ I
G g1 , g 2
∈ G
tl(g2 )
= x
2
ml(g1 ) ml(g2 )
G = g1 = x y y, g2 = y 2 I = g1, g2 g1 , g2 g3
tl(g3 )
= x
2
{
− x, g
· −
3
= x2 + x
}
tl(g1 )
= x y
·
G
I G
⊂K
[n]
G
⊂ I
H
⊂ I
I
H G
G
ml(hi )
= ml(gi )
i = 1, . . . , #G
H g1 h3 ml(H ) h3 = h 1 = h 2
∈ ml(G)
g1
H
∈G
g1 #G = #H
h1 H ml(h1 ) ml(g1 ) ml(g1 ) ml(g) ml(h) ml(g1 )
|
∈
#G = s < t = #H h1 h2 ml(H ) h1 h2
ml(g1 )
|
H
| ml(h ) 1
h
∈ H
g
|
ml(g1 )
g2 G
∈ G g = g
H
I
[n]
⊂ K
I G
2
G
∈ G
= ml(h1 )
1
⊂ I
G g
∈ G
h G
∈ \ {g}
rh (g) = g
G G
G
\{g}
K[n] G P = ml(g) + rG (g
I
{
I
\ {g} g ∈ G r (g) q , . . . , q − ∈ K
I
− ml(g)) | g ∈ G}
G
G
g G
H
1
m
∈ N
[n]
m 1
H = G
\ {g}
G
H
m
g g1 , g2 , . . . , gm−1
−1
m 1
−
g =
gi q i + rH (g),
·
i=1
rH (g) = ml(g) +rH (g g
− ml(g))
m 1
rH (g) = g
− −
gi q i =
ml(g) + rH (g
i=1
I H
rH (g
Q =
{ml(g) + r
ml(g)
H G
− ml(g))
− ml(g)) | g ∈ H },
H (g
Q Q = P
− ml(g)),
I
P
g rH (g
− ml(g))
I G = g1 = x y
I = g1 , g2
2
gi
2
3
G
gi
K[n]
I
I
{h , . . . , h } g ∈ G h g ≺ h 1
2
· − y, g = y − x, g = x +x}
{
⊂K
[n]
G = g1 , . . . , gk
{
k
ml(hi )
i
i
i
= ml(gi ), i N k gi H gi = h i
∈
rH (gi ) = 0
i
G
rgj (gi ) = g i
ml(H )
gi
ml(hi )
j = i
G j = i ml(g ) = ml(g ) | ml(h ) | ml(g ) ⇒ ml(h ) = ml(g ) = ml(h ), i∈N ml(h ) = ml(h ) i
p
p
H = G
K[n]
i
i
≺ h
i
ml(gi )
ml(g j )
j
gi = h i
gi
M(gi )
H
ml(h p )
H =
}
i
i
p
k
I, J G
J = G = I
[n]
⊆K
C[x,y,z ] z y
≺ ≺ x
x2 y2 z 2
− yz = 3, − xz = 4, − xy − 5. S = x2
2
2
{ − yz − 3, y − xz − 4, z − xy − 5}
13x + 11z = 0, 13y z = 0, 36z 2 = 169.
−
z
f f : Kn f ( p) = 0
→K
p
n
∈ K
[n]
∈K
f n
q
∈ K
C
C[z ] C C[n] f x1, . . . , xn−1
∈ C[x , . . . , x ] 1
xn x1 , . . . , xn−1
C[n]
p p q = 1
·
n
∈ C
[n]
degxn (f ) > 1 C
f
xn )
q C [n]
∈
p
deg( p) < 1 C q = p1 p C q p q = 1 p p(u) = 0 p(z ) q (z ) = 1 z Cn, p(z ) = 0 q (z ) = 0
p
∈ C
∈
p1 , p2 , . . . , pm C[n]
q 1 , . . . , qm
∈
[n]
s1 , . . . , sr
∈K
G p
q p = 1
∈
·
n
u
p
·
·
n
z
C[n]
∈ C
∈C p · q + ·· · + p · q
p
u
p
C[n]
[n]
1
1
m
S p I = s1, . . . , sr
m
=1
G = g1 , . . . , gt
{
}
G
S p n
z I
∈K
g= z
si (z ) r i=1 ai
·
= 0 i Nr si ai K[n] G p
∈
g(z ) =
∈
r i=1
S p g ai (z ) si (z ) = 0 G p
·
S p
I I
⊂ K[x , . . . , x ] 1
I
∩
I j
2
2
1
K[x, y] x, g3 = x 2
∩
2
y x
∈ N
≺ x G = {g = x · y − y, g = y − − } g , g g =0 {g , g } g (0, 0), (1, −1) (1, 1) 1
x=1
G I j = I K[x1 , . . . , xn− j ], j G j = G K[x1 , . . . , xn− j ] n
1
2
3
x =0
2
g1
S = x2 yz
2
2
− 3, y − xz − 4, z − xy − 5}
C[x,y,z ]
13x + 11z = 0, 13y z = 0, 36z 2 = 169.
−
z
≺ y ≺ x
{ −
z z =
13 6
z =
−
13 6
(
−
13x = 13y =
11 1 13 , , ) 6 6 6
13 6
−11 · (± ), ± . , − , − ) 13 6
( 11 6
1 6
13 6
C R C
f, f 1 , f 2 , . . . , fm s N 0
f
f , . . . , f ∈ C[x , . . . , x ] f ∈ I = f , . . . , f ∈ \{ } g = 1 − yf ∈ C[x , . . . , x , y] J = C[x , . . . , x ] ⊂ C[x , . . . , x , y] C = {f , . . . , f } T = C ∪ {g } α = (α , . . . , α , α ) ∈ C f (α) = f (α) = f (α) = 0 1 − α f (α) = g(α) = 1 h, h , . . . , h ∈ C[x , . . . , x , y] 1 = h(1 − yf ) + h f + ··· + h f
y
n
1
s
1
m
n
1
f
n
1
1
··· = f (α) = g(α) = 0 1 ∈ T m
m
m
1
f , . . . , f , g 1
1
n
1
m
n+1
y
n
1
2
y
1
m
1 1
1
n
m m
y = 1/f m
1=
hi (x1 , . . . , xn, 1/f )f i (x1 , . . . , xn, 1/f ).
i=1
s N 0 g1 , . . . , gm
∈ \
f s
∈ C[x , . . . , x ] 1
n
m
s
f =
gi (x1 , . . . , xn )f i (x1 , . . . , xn),
i=1
f s s
f 1, . . . , fm
f
∈ I
f
f 1
f 1 , . . . , fm
∈ f , . . . , f , 1 − yf 1
m
C[x1, . . . , xn ] C[x1 , . . . , xn ] γ i
∈ N \ {0}
xγ i i
∈ ml(I )
xn
≺ ·· · ≺ x
i
∈N
n
f 1 , . . . , fm
∈
1
I = f 1 , . . . , fm
i gn
∈ I
ml(gn )
gn gn−1 C[xn−1 , xn] gn−1 (xn−1, z n ) = 0
= gn = 0 xn
xn
≺ x −
∈
γ nγ n−1
··· = f
γ n−1
m
gn
∈
=0
∈N
n
∈ \ {0}
gn−1 gn = 0
∈ K, i ∈ N }
degxn (gn ) = γ n γ n− I ml(gn−1 ) = xn−1
1
−
·· ·
− ∈ {
···
⊂
∈
} ⊂
1
∈ N
f 1 , . . . , fn
∈ ∈
∈
Kn = (a1 , . . . , an )
n
n
n
∈
∈ ml(I )
gn = g n−1 = 0 i = 1, 2, . . . , n 2 g1 = g 2 = = g n = 0 γ 1 γ 2 . . . γn g1 = g 2 = = g n = 0 f 1, . . . , fm l1 = (z 11 , . . . , zn1 ), . . . , lk = z ij ) C[xi ] C[x1 , . . . , xn] i N n k N l1 , . . . , lk hi i Nn ksi s sn h1 , . . . , hn I ml(hi ) = x i ml(I )
K ai
xγ i i
γ i N C[xn ]
z n
k hi = j=1 (xi i s1 , . . . , sn
(z 1k , . . . , znk ) k ml(hi ) = xi
n
n 1
f 1 = f 2 =
i
xγ nn
∈ N
{
[n]
|
[n]
∈ K I ⊆ K V (I ) = {a ∈ K | f (a) = 0, ∀f ∈ I } I ⊂ K n
I
[n]
Kn
I
V (I )
I W (V (I )) = I
⊆ I (V (I ))
I
I
I
[n]
I (W ) = {f ∈ K | f (a) = 0, ∀a ∈ W } I (V (I )) I
G J J = G
G
x1 , x 2 , . . . , xn
B C[x1 , . . . , xn ]
G=0 G G V (J )
f x1 , x 2 , . . . , xn
G
R[n] f, g1 , g2 , . . . , g p , h1 , . . . , hq R[n] n,p,q N n n f : R R gi : R R i N p h j : Rn
∈
→
→
∈
∈
→ R j ∈ N
q
f (x1 , . . . , xn )
gi 0, i N p h j = 0, j Nq .
∈ ∈
gi
hi
I f (x) f (x) L < ε
|
⊂ R
x
z
− |
f, g : I
⊂ R → R
limx→z (f + g)(x) = limx→z f (x) + limx→z g(x) limx→z (f g)(x) = limx→z f (x) limx→z g(x)
·
·
f : I R ε > 0 δ > 0 L = limx→z f (x)
→
z I
∈
z
∈ I L ∈ R 0 < |x − z | < δ
I z
⊂R
f : I
f (x) x→z x lim f I
z
→R
− f (z ) . − z
z
∈ I
∂f (z) ∂x
f (z )
f
f : I f
→ R
f
∂ ∂x
f 2
∂ f ∂x 2
f ∂ n f ∂x n
a, b
∂ ∂x
∈R
nxn−1
f, g : I
⊂ R → R
∂ (a f + b g) ∂f ∂g = a + b . ∂x ∂x ∂x
·
∀n ∈ N
·
Dom(f )
f (x)
·
·
f (x) = a n xn +
f (x) = xn f (x) = f (x)
··· + a · x + a f (x) = n · a · x − + ·· · + 2 · a · x + a . f, g : R → R (f (g (x))) = f (g(x)) · g (x) f : I ⊂ R → R f x ∈ V x = a ·
n
f (a) a
f (n)
n
1
n 1
0
2
1
I m(g) a V
a
⊆
∈ I
I
f
2
∂ f (a) ∂x 2
f : I xi , i
a I
∈
∂ 2 f (a) ∂x 2
a >0
⊂ → R f : I
⊂ R → R ∂ 2 f (a) ∂x 2
a
<0
=0
n
⊂ R → R ∈ N n
i N m m Rm
∈
f : I R =0
∂f (a) ∂x
∈ N
x1 , . . . , xn z
∈ I
∂f (z) ∂x i
F : I F (x1, . . . , xn)
⊂
J F (z ) =
f i : I Rn R n R m z
⊂ →
∂f 1 (z) ∂x 1 ∂f m (z) ∂x 1
... ...
→R
∂f 1 (z) ∂x n ∂f m (z) ∂x n
z
,
z I (f 1 (z ), . . . , fm (z ))
∈
∈
n
f : I f
z
∇f (z )
⊂ R → R
z I f
z
∈
∂f (z) ∂x 1
∇f (z ) =
f : I
,
∂f (z) ∂x n
⊂ R → R
z
∈ I
f
2
∇ f (z ) = n
∈ N
∂ 2 f (z) ∂x 21 2 ∂ f (z) ∂x 2 x1
...
∂ 2 f (z) ∂x n x1
...
...
∂ 2 f (z) ∂x 1 xn ∂ 2 f (z) ∂x 2 xn ∂ 2 f (z) ∂x 2n
M = (aij )
.
∈ M (R) n
M
a11
2
∇ f
f (x) h(x) = 0, x
n
f : Rn
∈R
S = x
→R n
h = (h1 , . . . , hq ) : Rn
{ ∈ R | h (x) = 0, ∀i ∈ N } i
q
→R
∈ N
S
q
x : I (x1(t), . . . , xn (t)) (x1(t), . . . , xn (t))
q
∈ S
t
R
⊂
∈ I
→
S t
∈ I
x S t I h(x(t)) = 0
x : I
→ S
∈
x(t) = x (t) = S
x
∈
J h (x(t))x (t) = 0
z z
S
x
n
T = y
{ ∈ R |
x J h (x)y = 0
S
}
T
n
z
∈ R
Ker(J h (z )) f (x)
S
Ax = b A
A Ker(A)
Ax = 0 T
z
λ1 , . . . , λq
∈ R
q
∇f (z ) =
i=1
λi
· ∇h (z ). i
x : I φ(s) = f (x(s))
⊂
R
→
S
z = x(s )
s
s φ (s) = J f (x(s))x (s) = z z λ
∈R
q
f f (z ) Ker (J h (z ))
∇
∇f (z ) ∇f (z ) = J (z ) · λ t h
t
∇ f (x(s))x(s). T f (z )
∇
∈
S Im(J h(z ))
λi
L(x, λ) = f (x) + λt h(x)
·
f (x) + J ht (x)
∇ L(x, λ) = ∇ x
J Lt (x, λ) = J f t (x) + J ht (x) λ x L(x, λ) = 0
·λ
∇ L(x, λ) ∇ L(x, λ)
·
∇
x
2 x
2 x
∇ L(x, λ)
z
λ t h
yt
∇f (z ) + J (z ) · λ = 0 T
2 x
· ∇ L(z, λ) · y 0
S
z
x : I s
⊂ R → S
φ(s) = f (x(s)) t
φ (s ) = J f (x(s )) x (s ) =
λ
z = x(s )
∇ f (x(s )) · x(s )).
·
φ (s) = x (s )t
q
∈R y ∈ T ,
2
t
· ∇ f (x(s )) · x(s ) + ∇ f (x(s )) · x(s ). ψ (s) = λ · h(x(s)) z λ · h(x(s )) = 0
m
t
∈ R
t
q
ψ (s ) =
λi
i=1
t
· ∇ h (x(s )) · x(s ) = 0, i
q
ψ (s ) =
i=1
λi [x (s )
·
2
t
· ∇ h (x(s )) · x(s )] + ∇ h (x(s )) · x(s )] = 0. i
i
q
φ (s ) = x (s )t · [∇2 f (x(s )) +
i=1
q
λi
·∇
2
hi (x(s ))] · x (s ) + [∇f (x(s )) +
i=1
λi
t
·∇h (x(s ))] · x(s ).
z
i
λ q
φ (s ) = x (s )t · [∇2 f (x(s )) +
λi
i=1
2
· ∇ h (x(s ))] · x(s ) 0, i
x (s )
s T
q
2 x
2
y
2
∇ L(z, λ) = ∇ f (z ) + λ · ∇ h (z ), y ∈ T . · ∇ L(z, λ) · y 0 i=1
t
2 z
i
i
z λ
∈R
S T
S
z
q t h
∇f (z ) + J (z ) · λ = 0
yt
2 z
· ∇ L(z, λ) · y > 0
y
∈ T \ {0},
z
f (x)
n
f : I
⊂ R → R
hi : I
→ R, i ∈ N
p
g j
g j (x) 0, j hi (x) = 0, i
∈ N , ∈N , : I → R, j ∈ N q
p
q
n,p,q N
∈
f z z
∇f (z )
z
z
g j (x) 0
hi (x) = 0, i
g := (g1 , . . . , gq ) : I
λ
q
∈ R ,µ ∈
→
∈
z N p , j Nq
z S = x
z z
∈ }
z I Rq
W g (z ) z
∈
z s(z ) Rs(z)
{ ∈ I |
h := (h1 , . . . , h p ) : I R p z t t f (z ) + J h (z ) λ + W g (z ) µ = 0 µk
∇
→
·
·
k
∈ N
0
s(z)
µ µ t h
t g
∇f (z ) + J (z ) · λ + J (z ) · µ = 0,
p
∈ R
µi
µi gi (z ) = 0,
·
N p
µi
i
0,
∈N . p
L(x,λ,µ) = f (x) + λ t h(x) + µ t g(x) L(z,λ,µ) = 0 µi gi (z ) = 0,
∇
z
µi
·
z
0 yt
∈
q
λ
∈ R
t h
µk
i
0,
k
µ
∈ R
s(z)
t g
∇f (z ) + J (z ) · λ + W (z ) · µ = 0
∈ N · ∇ L(z,λ,µ) · y 0, s(z)
2 z
n
y
∈ T = {y ∈ R | J (z ) · y = 0
W g (z ) y = 0 ,
·
h
}
W g
∇ L(x,λ,µ) = ∇ f (x) + 2 x
2
q i=1
λi
∇ h (x) + 2
p j=1 µ j
i
2
∇ g (x) j
z S µi
i
0
∈N
λ
∈
s(z )
t h
t g
∇f (z ) + J (z ) · λ + W (z ) · µ = 0 yt W g
T
2 z
· ∇ L(z,λ,µ) · y > 0,
y
z
∈ T ,
f, g1 , g2 , . . . , g p , h1, . . . , hq R[n] n,p,q N n n n f : R R gi : R R i N p h j : R R j Nq
→
→
∈
∈
∈ → ∈
f (x1 , . . . , xn )
gi 0, i N p , h j = 0, j Nq .
∈ ∈
∈R
q
µ
s(z)
∈ R
n,p,q
∈ N
f, g1 , g2 , . . . , g p , h1 , . . . , hq
[n+ p]
∈ R
f (x1 , . . . , xn , . . . , xn+ p ),
C
g1 + x2n+1 =: g 1 = 0, ... g p + x2n+ p =: g p = 0, h j = 0, j Nq .
∈
R
∇
f (z ) + J gt (z )λ + J ht (z )µ = 0, µi gi (z ) = 0, g (z ) = 0, h(z ) = 0,
z = (x1 , . . . , xn , . . . , xn+ p )
Rn+ p λ
q
p
∈ R µ ∈ R
λ µ C R
\
f (z ) = 9x21 + x22 + 9x23
g1(z ) = 1 x1 x2 0, g2(z ) = 1 x2 0, g3(z ) = x 3 1 0.
− − −
y1 , y2 , y3
z 1
18x1 y1 x2 = 0, 2x2 y1 x1 y2 = 0, 18x3 + y3 = 0, y1 (1 x1 x2 ) = 0, y2 (1 x2 ) = 0, y3 (x3 1) = 0, 1 x1 x2 + z 12 = 0, 1 x2 + z 22 = 0, x3 1 + z 32 = 0.
3
1
2
3
1
2
− − − − − −
− − −
≺ z ≺ z ≺ y ≺ y ≺ y ≺ x ≺ x ≺ x 2
z 1 , z 2 , z 3
J
⊂ C[x , x , x , y , y , y , z , z , z ] 1
2
3
− − − − −− − − −− − − − − −
18x3 + y3 = 0, 18 y3 + 18z 32 = 0, 1 x2 + z 22 = 0, y32 + 18y3 = 0, 2y2 + y22 + y2y1 = 0, 36 + y12 + 2y2 y22 36z 12 = 0, y2 + y2x2 = 0, y1 x2 18x1 = 0, 2x22 y2 y1 = 0, 18y2x1 2y2 + y22 = 0, 2x2 + y1 x1 + y2 = 0, 1 + x1 x2 z 12 = 0, 18x21 y1 = 0, 16y2 + y22 + 18y2 z 12 = 0, 32y2 + 14y22 + y23 = 0, y1 z 12 = 0, 16y2 + y22 + 36x2 z 12 = 0, x1 z 12 = 0, z 12 + z 14 = 0.
− −
√ √ √ p = ( 3/3, 3, 0, 6, 0, 0, 0, 3 − 1, 1) √ √ √3 − 1, −1) p = ( 3/3, 3, 0, 6, 0, 0, 0, √ √ √ 3 − 1, 0) p = ( 3/3, 3, 1, 6, 0, −18, 0, p = (1, 1, 0, 18, −16, 0, 0, 0, 1) p = (1, 1, 0, 18, −16, 0, 0, 0, −1) p = (1, 1, 1, 18, −16, −18, 0, 0, 0) 1
2
3
4
5
6
3
1
2
3
1
2
3
p √ p √ √ √ ( 3/3, 3, 0), ( 3/3, 3, 1), (1, 1, 0) 1
p4
2
(1, 1, 1)
√ √ 3, 0)
z = ( 3/3,
f (z ) = 6
I
1
ml(I )
f (z ) = x 21 + x22 + x23
g1 (z ) = x 21 + x22 + x23 4 = 0, g2 (z ) = x 21 + x22 (x3 1)2 = 0.
− − −
(0, 0, 1) 2
(0, 0, 0)
∈ g , g
1
1
2
y1 , y2
∈ R
p5
2x1 (y1 + y2 + 1) = 0, 2x2 (y1 + y2 + 1) = 0, 2x3 (y1 y2 + 1) 2y2 = 0, x21 + x22 + x23 4 = 0, x21 + x22 (x3 1)2 = 0.
− − − − −
C[x1 , x2 , x3 , y1 , y2 ]
y1
≺ y ≺ x ≺ x ≺ x 2
1 + y1 = 0, y2 = 0, x3 + x21 + x22 = 5/2, x23 x3 = 3/2.
−
1
2
3
√ = (1 ± 7)/2
x3 (−) x3
R3 z
xy
x(+) 3
S := {(x , x , (1 − √ 7)/2) | x + x = 2 + √ 7/2} 1
2 1
2
2 2
S
g12 (z ) =
x21
x 22
x 23 +
− − −
4
g1 (z ) = x21 + x22 + x23 0
g11 (z ) = x21 + x22 + x23
− 4 = 0
− 4 0
y3 z 1 , z 2
− −
2x1 (y1 + y2 y3 + 1) = 0, 2x2 (y1 + y2 y3 + 1) = 0, 2x3 (y1 y2 y3 + 1) 2y2 = 0, y1 (x21 + x22 + x23 4) = 0, y3 (x21 + x22 + x23 4) = 0, x21 + x22 + x23 4 + z 12 = 0, x21 x22 x23 + 4 + z 22 = 0, x21 + x22 (x3 1)2 = 0.
− − − −
− − − − −
− −
J y1
≺ y ≺ y ≺ x ≺ x ≺ x 2
3
1
2
−
⊂ C[x , x , x , y , y , y , z , z ] 1
2
3
1
2
3
− − − − −− − − − − − − − − −− − − − − − − − − − − − − −−
z 12 + z 22 = 0, y1 y2 = 0, 4y2 + 12y22 11 11y1 + 11y3 + 2z 12 = 0, y1 y12 + y1y3 = 0, y2 y3 = 0, y1 y3 + y32 y12, x1 y1 x1 y2 + x1 y3 x1 = 0, x2 y1 x2 y2 + x2 y3 x2 = 0, 2x21 + 2x22 5 + z 12 + 2x3 = 0, 4 + z 12 y2 + 3x3 y2 4y1 + 4y3 = 0, 3x3 + 4y2 3x3y1 + 4 z 12 + 4y1 4y3 + 3x3 y3 = 0, 3 2x3 + 2x23 + z 12 = 0, y1 z 12 = 0, y3 + 1 z 12 5y2 + y1 + 3y2 z 12 = 0, y3 z 12 = 0, 3x1 y2 + 2x1 z 12 = 0, 3x2 y2 + 2x2 z 12 = 0, 13 13y1 + 13y3 16y2 + z 12 + 6x3 z 12 = 0, 47 + 47y1 47y3 35z 12 + 8y2 + 6z 14 = 0. V (J )
−
3
1
2
z 1
≺ z ≺ 2
− − −− − − − − − − − − − − − − − − − − − − −
z 1 = 0, y12 + y32 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 + 2x22 = 0, z 2 = 0.
z 1 = 0, y1 = 0, y3 + 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 + 2x22 = 0, z 2 = 0.
z 1 = 0, y1 = 0, y3 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 + 2x22 = 0, z 2 = 0.
z 1 = 0, y12 + 1 = 0, y3 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 + 2x22 = 0, z 2 = 0.
z 1 = 0, 2y12 + 1 = 0, 1 + 2y32 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 + 2x22 = 0, z 2 = 0. z 1 = 0, y12 + y32 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, x1 = 0, 5 + 2x3 + 2x22 = 0, z 2 = 0.
−
− − − − − − − − − −− − − − − − − −− −
z 1 = 0, y12 + y32 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 = 0, x2 = 0, z 2 = 0.
−
2z 12 3 = 0, y1 = 0, y3 = 0, 1 + y22 = 0, 1 + 2x3 = 0, 9 + 4x21 + 4x22 = 0, 3 + 2z 22 = 0.
z 1 = 0, y1 = 0, y3 + 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, x1 = 0, 5 + 2x3 + 2x22 = 0, z 2 = 0. z 1 = 0, y1 = 0, y3 + 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 = 0, x2 = 0, z 2 = 0. z 1 = 0, y1 = 0, y3 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, x1 = 0, 5 + 2x3 + 2x22 = 0, z 2 = 0. z 1 = 0, y1 = 0, y3 1 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 = 0, x2 = 0, z 2 = 0.
− − − −− − − − − − − −− − − − −
z 1 = 0, y12 + 1 = 0, y3 = 0, y2 = 0, 3 2x3 + 2x23 = 0, x1 = 0, 5 + 2x3 + 2x22 = 0, z 2 = 0. z 1 = 0, y12 + 1 = 0, y3 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 = 0, x2 = 0, z 2 = 0. z 1 = 0, 2y12 + 1 = 0, 1 + 2y32 = 0, y2 = 0, 3 2x3 + 2x23 = 0, x1 = 0, 5 + 2x3 + 2x22 = 0, z 2 = 0. z 1 = 0, 2y12 + 1 = 0, 1 + 2y32 = 0, y2 = 0, 3 2x3 + 2x23 = 0, 5 + 2x3 + 2x21 = 0, x2 = 0, z 2 = 0. 3 + z 12 = 0, y1 = 0, y3 = 0, 1 + 2y22 = 0, x3 1 = 0, x1 = 0, x2 = 0, 3 + z 22 = 0.
x23 x3 = 3/2, x3 + x21 + x22 = 5/2,
−
2z 12 3 = 0, y1 = 0, y3 = 0, 1 + y22 = 0, 1 + 2x3 = 0, x1 = 0, 2x2 3 = 0, 3 + 2z 22 = 0.
−
−
2z 12 3 = 0, y1 = 0, y3 = 0, 1 + y22 = 0, 1 + 2x3 = 0, x1 = 0, 2x2 + 3 = 0, 3 + 2z 22 = 0.
−
2z 12 3 = 0, y1 = 0, y3 = 0, 1 + y22 = 0, 1 + 2x3 = 0, 2x1 3 = 0, x2 = 0, 3 + 2z 22 = 0.
−
−
2z 12 3 = 0, y1 = 0, y3 = 0, 1 + y22 = 0, 1 + 2x3 = 0, 2x1 + 3 = 0, x2 = 0, 3 + 2z 22 = 0.
−
f (z ) S g (x)
H G
f
I
S g
H G
∩
G I
⊂ J
x3 = (1
− √ 7)/2
x21 + x22 = 2 +
J
√ 7/2 (1, 0, 0)
g(z ) = (x1 1)2 + x22 + x23 , f (z ) = x 21 + x22 + x23 ,
−
g1 (z ) = x 21 + x22 + x23 4 = 0, g2 (z ) = x 21 + x22 (x3 1)2 = 0,
− − −
f (z )
g (z ) x21 + x22 = 2 +
g (z )
√ 7/2, x
3
= (1
f (x)
y1 , y 2
2(x1 1) + 2x1 y1 = 0, 2x2 (1 + y1 ) = 0, y2 = 0, x21 + x22 = 2 + 7/2, x3 = (1 7)/2,
−
− √
√
C[x1 , x2 , x3 , y1 , y2 ] x3
− √ 7)/2
y1
≺ y ≺ x ≺ x ≺ 2
1
2
√ (1/3) 7, 0)
y2 = 0, 2x1 + ( 4 7)y1 = 4 + x2 = 0, 2x3 = 1 7,√ 7−1 2 2y1 + y1 = , 9
− − √ − √ −
√ 7,
√ √ z = (1/2+(1/2) 7, 0, 1/2 − (1/2) 7, −4/3+ √ √ √ = (−1/2 − (1/2) 7, 0, 1/2 − (1/2) 7, −2/3 − (1/3) 7, 0) 1
z 2
g(z 1) = 0.11992
g(z 2 ) = 7.4116
z 1
f : N A
→ R
t:N A
→ R
O(f (x))
A
f (x) t O(f (x)) = g : N
∈
{
→ R | ∃r, N ∈ N, k N → rf (k) g(k)},
f
t
d0 > 0, d1 > 1, . . . , d p > 1 p I = q, f 1 , . . . , f p O(log(A)/log(G)) δ log(A) (n) A log(G)
∈
q, f 1 , . . . , f p N 0 < p < n
[n]
∈Q
δ G
{d , d , . . . , d , max (d − 1), . . . , max (d − 1)} 0
1
p
1i p
i
1i p
p
δ = (
i=1
di )
n p
−
(d0
i0 + +ip =n p
···
−
− 1)
i
i0
. . . (d p
ip
− 1)
I
I
Q
δ
n δ D (D − 1) − p
n p
p
(2D)n = D O(n) ,
D = max deg(f 1 ), . . . , deg(f p ),deg(q )
{
DO(n) .
}
D >1
hi =
n j=1
aij x j
d, h1, . . . , hq Z[x1 , . . . , xn ] ci N 0 i Nq j Nn 0
∈ ∈ ∪ { } ∈ ∈ ∪ { }
aij
d(x1 , . . . , xn ) = b 1 x1 +
n j=1
hi =
0
∈ Z
n, q
∈ N j ∈ N
q
··· + b x , n n
aij x j = c i , i
∈N . q
K[y1 , . . . , yq ] K n
yi i
∈ N
j =1 aij xj
= y ici ,
q n
y1
j =1 a1j xj
n
. . . yq
j =1 a qj xj
= y 1c . . . yqcq . 1
x1 , . . . , xn x j , j
∈ N
n
(y1a y2a . . . yqaq )x . . . (y1a n y2a n . . . yqaqn )xn = y 1c . . . yqcq . 11
21
1
1
1
2
φ : K[z 1 , . . . , zn ]
1
→ K[y , . . . , y ] 1
q
Im(φ) = K[m1 , . . . , mn ]
a
φ(z i ) = m i = y 1a i . . . yq qi 1
c
y1c . . . yqq 1
Zn0 c y1c . . . yqq = φ(z 1s . . . zns n )
∈ I m(φ)
1
x = (s1 , . . . , sn )
1
n
(v1 , . . . , vn )
∈ Z
0
(y1a y2a . . . yqaq )v . . . (y1a n y2a n . . . yqaqn )vn = y 1c . . . yqcq 11
21
1
1
1
2
1
φ (φ(z 1 ))v . . . (φ(z n))vn = φ(z 1v ) . . . φ(z nvn ) = φ(z 1v . . . znv n ) = y 1c . . . yqcq 1
1
1
c
z 1v . . . znv n K[z 1 , . . . , zn ] y1c . . . yqq c u K [z 1 , . . . , zn ] φ(u) = y 1c . . . yqq u = z 1s . . . zns n si Z0 i Nn
∈
1
∈
1
∈ Im(φ) u
1
∈
1
1
∈
φ(u) = φ(z 1s ) . . . φ(z nsn ) = (y1a y2a . . . yqaq )s . . . (y1a n y2a n . . . yqaqn )sn = y 1c . . . yqcq 1
11
21
1
2
1
n
(s1 , . . . , sn )
∈Z
0
c
y1c . . . yqq 1
φ(h)
1
1
c = y 1c1 . . . yqq
∈ I m(φ)
h
∈ K[z , . . . , z ] 1
n
Im(φ) K[z 1, . . . , zn ]/Ker(φ) Ker(φ) κ : K[z 1 , . . . , zn ]/Ker(φ) a
∈ K[z , . . . , z ] 1
κ(a + Ker(φ)) = φ(a)
→ I m(φ)
n
Ker (φ) Ker(φ) aα1 1
Z0
mi
A . . . aαnn
bα1 1
−
. . . bαnn
∈ −
−
− ∈
q
−
z i , i
G = G
∩ K[z , . . . , z ] 1
n
f I m(φ)
∈
s i=1 gi ti
f = h + G rG (f ) = h
G
h + Ker(φ) G = g1 , . . . , gs J ti
J
∈ K[y , . . . , y ] 1
q
α1 , . . . , αn
∈
{
}
Im(φ)
∩
G J y j , j Nq Ker(φ) φ
∈ N
∈
n
h
J f
∈ A
J = z 1 m1, . . . , zn mn K[z 1 , . . . , zn , y1 , . . . , yq ] i Nn Ker (φ) = J K[z 1 , . . . , zn ]
∈ K[y , . . . , y ] 1
a1 , . . . , an , b1, . . . , bn a1 b1 , . . . , an bn
∈ K[z , . . . , z ] 1
n
h
∈
φ(h) = f Ker(φ) Ker(φ)
∈ K[z , . . . , z ], i ∈ N 1
n
s
⊂ K[z , . . . , z , y , . . . , y ] h ∈ K[z , . . . , z ] 1
n
1
q
1
n
z y rG (f ) = h
≺
f = 0
G
z
≺ y
h=
J
c rG(y1c1 . . . yqq )
h K[z 1 , . . . , zn ] h = z 1x . . . znx n
∈
1
(x1, . . . , xn )
y1x
1
x1 + x2 + x3 = 3, 3x1 + 5x2 + 2x3 = 10.
+x2 +x3 3x1 +5x2 +2x3 y2
= y 13 y210 ,
(y1 y23)x (y1 y25 )x (y1 y22 )x = y 13y210 . 1
m1 = y 1 y23, m2 = y 1 y25 J G = z 3 y2
{
2
m3 = y 1 y22
3
J = z 1
3 1 2
5 1 2
2 1 2
− y y , z − y y , z − y y 2
3
z 3
2 3 2
3 1
2 1 2
2 1 2
3 3
2 1 1
≺ z ≺ z ≺ y ≺ y 2
1
2 3
2
1
2 1 2
− z , −z z + z y , −z z + z , z y − z z , z y − z , −z + z y , z y y − z , −z + y y }. 1
3 1
2 1
rG(y13 y210 )
3 2
2
1 1 2
3
y13 y210 = z 3 z 1 z 2
G
(x1 , x2 , x3 ) = (1, 1, 1)
z y
≺
d(x1, x2 , x3 ) = 3x1 + x2 + 2x3
y1x
1
x1 + x2 + x3 = 4, x1 + x2 + 2x3 = 5.
+x2 +x3 x1 +x2 +2x3 y2
= y 14 y25 ,
(y1y2)x (y1 y2 )x (y1y22 )x = y 14 y25 . 1
2
3
z 3 y2 y 1 (0, 3, 1)
≺
rG (y14 y25)
= z 3 z 23
{ − z , y z − z , − −z + y y } z z H = {−z + z , y z − z , z y − z , −z + y y } r (y y ) = z z
G = z 1
2
2 2
3
2
2 1
3
1
(3, 0, 1)
y1 z 3 + z 22 ,
2
1 3
2 1
1 2
3
1
2 1
≺ z ≺ z ≺ 2
1
(x1 , x2 , x3 ) =
1
H
4 5 1 2
3 1 3
(x1 , x2 , x3 ) =
(3, 0, 1) d(3, 0, 1) = 11
d(0, 3, 1) = 5
Nn d(x1, . . . , xn) < d(x1 , . . . , xn )
z 1x1
c . . . znx n
≺
d x1 φ(z 1 . . . znx n )
φ . . . znx n
x1 d z 1
≺
=
x φ(z 1 1
z i , i x . . . zn n )
∈
G J
≺ z ≺ y d
φ (x1, . . . , xn )
z
rG (z 1c . . . znc n ) = z 1x . . . zn xn 1
1
x
J z 2 y 2 y 1 6 d(2, 1, 1) = 8
≺ ≺
(3, 0, 1)
z 1 z 3 d(0, 3, 1) = 5, d(1, 2, 1) =
≺ ≺
• • •
