Hình học đại số tính toán pot

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Phạm Tiến Sơn
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Hình học đại số tính toán 1







Đà Lạt - 2008
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k[x
1
, x
2
, . . . , x
n
].
x
1
, x

2
, . . . , x
n
x
α
1
1
x
α
2
2
. . . x
α
n
n
,
α
1
, α
2
, . . . , α
n
α
1
+ α
2
+ ··· + α
n
α = (α
1

, α
2
, . . . , α
n
),
|α| = α
1
+ α
2
+ ··· + α
n
,
x
α
= x
α
1
1
x
α
2
2
. . . x
α
n
n
.
f x
1
, x

2
, . . . , x
n
k
k;
f =

α∈Λ
a
α
x
α
, a
α
∈ k,
Λ N
n
. k[x
1
, x
2
, . . . , x
n
]
x
1
, x
2
, . . . , x
n

k.
1, 2, 3 k[x], k[x, y] k[x, y, z].
f(x, y, z) = 2x
3
yz
2
+ y
3
z
3
−
2
3
xyz
Q[x, y, z].
f =

α∈Λ
a
α
x
α
k[x
1
, x
2
, . . . , x
n
].
a

α
x
α
.
a
α
= 0 a
α
x
α
f.
f, deg f, |α| a
α
= 0.
f(x, y, z) = 2x
3
y
2
z + 5xy
3
+ 7xyz + 9z
3
∈ Q[x, y, z]. deg f = 6.
g
f h ∈ k[x
1
, x
2
, . . . , x
n

] g = f · h.
k[x
1
, x
2
, . . . , x
n
]
k[x
1
, x
2
, . . . , x
n
]
k n
k
n
= {(a
1
, a
2
, . . . , a
n
) | a
i
∈ k, i = 1, 2, . . . , n}
n k.
n = 1 k
1

n = 2 k
2
f =

α
c
α
x
α
∈ k[x
1
, x
2
, . . . , x
n
]
f : k
n
→ k, (a
1
, a
2
, . . . , a
n
) →

α
c
α
a

α
1
1
a
α
2
2
. . . a
α
n
n
.
k f = 0 k[x
1
, x
2
, . . . , x
n
]
f : k
n
→ k
k f = g k[x
1
, x
2
, . . . , x
n
]
f, g : k

n
→ k
C
f C.
a ∈ C f(a) = 0.
k k[x]
k.
R x
2
+ 1
R. C
p Z ≡
m ≡ n mod p
m − n p.
≡ F
p
F
p
p
F
p
\ {0}
a
p−1
= 1 a ∈ F
p
\ {0}.
a
p
= a a ∈ F

p
.
f ∈ F
p
[x] f F
p
.
f ∈ C[x
1
, x
2
, . . . , x
n
] Z
n
f ≡ 0.
f ∈ C[x
1
, x
2
, . . . , x
n
] M = deg
x
1
f.
Z
n
M+1
= {(x

1
, x
2
, . . . , x
n
) ∈ Z
n
| 1 ≤ x
i
≤ M + 1}.
f Z
n
M+1
f ≡ 0.
k f
1
, f
2
, . . . , f
s
∈ k[x
1
, x
2
, . . . , x
n
].
V (f
1
, f

2
, . . . , f
s
) = {(a
1
, a
2
, . . . , a
n
) ∈ k
n
| f
i
(a
1
, a
2
, . . . , a
n
) = 0, i = 1, 2, . . . , s}
f
1
, f
2
, . . . , f
s
.
R
2
V (x

2
+ y
2
−1)
f
V (y −x
2
, z −x
3
) R
3
.
a
11
x
1
+ ··· + a
1n
x
n
= b
1
,
a
21
x
1
+ ··· + a
2n
x

n
= b
2
,
=
a
m1
x
1
+ ··· + a
mn
x
n
= b
m
k
n
V, W ⊂ k
n
V ∪W V ∩W
V (z) ∪ V (x, y) = V (zx, zy).
R
2
,
V (x
2
+ 4y
2
+ 2x − 16y + 1).
V (x

2
− y
2
).
V (2x + y −1, 3x − y + 2).
V (y
2
− x(x − 1)(x − 2)).
R
2
,
V (x
2
+ y
2
− 4) ∩ V (xy −1) = V (x
2
+ y
2
− 4, xy −1).
R
3
,
V (x
2
+ y
2
+ z
2
− 1).

V (x
2
+ y
2
− 1).
V (x + 2, y −1.5z).
V (xz
2
− xy).
V (x
2
+ y
2
+ z
2
− 1, x
2
+ y
2
+ (z −1)
2
− 1).
k
n
(p
1
, p
2
, . . . , p
n

) ∈ k
n
k
n
{(x, x) ∈ R
2
| x = 1}
R
2
.
{(x, y) ∈ R
2
| y = 0}
R
2
.
Z
n
Z
n
.
V ⊂ k
m
W ⊂ k
n
V × W
R
3
x + y + z = 1,
2 + 2y −z = 3.

z = t,
x = −1 −2t,
z = 2 + 2t,
t ∈ R;
V (x
2
+ y
2
− 1) ⊂ R
2
x =
1 − t
2
1 + t
2
,
y =
2t
1 + t
2
,
t ∈ R. x =
1−t
2
1+t
2
t (−1, 0)
k f/g f, g ∈ k[t
1
, t

2
, . . . , t
m
]
g 0 t
1
, t
2
, . . . , t
m
k. f/g p/q qf = pg k[t
1
, t
2
, . . . , t
m
].
t
1
, t
2
, . . . , t
m
k k(t
1
, t
2
, . . . , t
m
).

k(t
1
, t
2
, . . . , t
m
)
V = V (f
1
, f
2
, . . . , f
s
) ⊂ k
n
.
f
1
= f
2
= ··· = f
s
= 0
V.
r
1
, r
2
, . . . , r
n

∈ k(t
1
, t
2
, . . . , t
m
) (x
1
, x
2
, . . . , x
n
)
x
1
= r
1
(t
1
, t
2
, . . . , t
m
),
x
2
= r
2
(t
1

, t
2
, . . . , t
m
),
=
x
n
= r
n
(t
1
, t
2
, . . . , t
m
),
V. r
1
, r
2
, . . . , r
n
V.
r
1
, r
2
, . . . , r
n

V.
V (x
2
− y
2
z
2
+ z
3
)
x = t(u
2
− t
2
),
y = u,
z = u
2
− t
2
,
u, t ∈ k.
V
V p ∈ k
n
V
•
• V.
V.
x = 1 + t,

y = 1 + t
2
.
V (y −x
2
+ 2x + 2).
x
2
+ y
2
= 1
x =
1 − t
2
1 + t
2
,
y =
2t
1 + t
2
.
V (y −x
2
, z −x
3
)
x = t,
y = t
2

,
z = t
3
.
x + 2y −2z + w = −1,
x + y + z −w = 2.
f ∈ k[x]. V (y −f(x)).
x =
t
1 + t
,
y = 1 −
1
t
2
.
(1, 1).
x
2
− y
2
= 1.
x = cosh(t),
y = sinh(t)
x
2
− y
2
= 1.
3

t.
x
2
+ y
2
+ z
2
−1 = 0
x =
2u
u
2
+ v
2
+ 1
,
y =
2v
u
2
+ v
2
+ 1
,
z =
u
2
+ v
2
− 1

u
2
+ v
2
+ 1
.
(n − 1)
x
2
1
+ x
2
2
+ ··· + x
2
n
= 1.
c C = V (y
2
− cx
2
+ x
3
).
C 3
y = mx, m = 0, C \{(0, 0)}
m
2
= c.
(1, t) ∈ V (x − 1) L (1, t)

(0, 0). L C (x, y).
C
x = c −t
2
,
y = t(c −t
2
).
I ⊂ k[x
1
, x
2
, . . . , x
n
]. I
0 ∈ I.
f, g ∈ I f + g ∈ I.
f ∈ I g ∈ k[x
1
, x
2
, . . . , x
n
] fg ∈ I.
{xf + y
2
g | f, g ∈ k[x, y]}
k[x, y].
{x
1

f
1
+ x
2
f
2
+ ··· + x
n
f
n
| f
1
, f
2
, . . . , f
n
∈ k[x
1
, x
2
, . . . , x
n
]}
k[x
1
, x
2
, . . . , x
n
].

f
1
, f
2
, . . . , f
s
∈ k[x
1
, x
2
, . . . , x
n
].
f
1
, f
2
, . . . , f
s
 =

s

i=1
f
i
g
i
| g
1

, g
2
, . . . , g
s
∈ k[x
1
, x
2
, . . . , x
n
]

.
f
1
, f
2
, . . . , f
s
 k[x
1
, x
2
, . . . , x
n
]. f
1
, f
2
, . . . , f

s

f
1
, f
2
, . . . , f
s
.
I ⊂ k[x
1
, x
2
, . . . , x
n
] f
1
, f
2
, . . . , f
s
I = f
1
, f
2
, . . . , f
s
; f
1
, f

2
, . . . , f
s
I.
{xf + y
2
g | f, g ∈ k[x, y]}
x y
2
.
x
1
, x
2
, . . . , x
n
{x
1
f
1
+ x
2
f
2
+ ··· + x
n
f
n
| f
1

, f
2
, . . . , f
n
∈ k[x
1
, x
2
, . . . , x
n
]}.
k[x
1
, x
2
, . . . , x
n
]
f
1
, f
2
, . . . , f
s
g
1
, g
2
, . . . , g
t

I
k[x
1
, x
2
, . . . , x
n
]. V (f
1
, f
2
, . . . , f
s
) = V (g
1
, g
2
, . . . , g
t
).
V ⊂ k
n
I(V ) = {f ∈ k[x
1
, x
2
, . . . , x
n
] | f(a
1

, a
2
, . . . , a
n
) = 0 (a
1
, a
2
, . . . , a
n
) ∈ V }.
I(V )
V = {(0, 0)} ⊂ k
2
.
I(V ) = x, y.
k
I(k
n
) = {0}.
V = V (y −x
2
, z −x
3
) ⊂ R
3
.
I(V ) = y −x
2
, z −x

3
.
I(V ) ⊃ y −x
2
, z −x
3

x
α
y
β
z
γ
= x
α
(x
2
+ (y −x
2
))
β
(x
3
+ (z −x
3
))
γ
= x
α
(x

2β
+ g
1
(y −x
2
))(x
3γ
+ g
2
(z −x
3
))
= h
1
(y −x
2
) + h
2
(z −x
3
) + x
α+2β+γ
= h
1
(y −x
2
) + h
2
(z −x
3

) + r,
g
1
∈ R[x, y], g
2
∈ R[x, z], h
1
, h
2
∈ R[x, y, z] r ∈ R[x].
f ∈ R[x, y, z] R
f = h
1
(y −x
2
) + h
2
(z −x
3
) + r
h
1
, h
2
∈ R[x, y, z] r ∈ R[x]. f ∈ I(V )
0 = f(t, t
2
, t
3
) = 0 + 0 + r(t)

t ∈ R. r ≡ 0. f ∈ y −x
2
, z −x
3
.
f
1
, f
2
, . . . , f
s
∈ k[x
1
, x
2
, . . . , x
n
].
f
1
, f
2
, . . . , f
s
 ⊂ I(V (f
1
, f
2
, . . . , f
s

)).
I(V (x
2
, y
2
)) = x, y. x ∈ x
2
, y
2
.
x
2
, y
2
 ⊂ I(V (x
2
, y
2
)).
V W k
n
.
V ⊂ W I(V ) ⊃ I(W ).
V = W I(V ) = I(W ).
x
2
+ y
2
− 1 = 0,
xy −1 = 0.

y
x
2
+ y
2
− 1, xy −1.
I ⊂ k[x
1
, x
2
, . . . , x
n
] f
1
, f
2
, . . . , f
s
∈ k[x
1
, x
2
, . . . , x
n
].
f
1
, f
2
, . . . , f

s
∈ I.
f
1
, f
2
, . . . , f
s
 ⊂ I.
x + y, x − y = x, y.
x + xy, y + xy, x
2
, y
2
 = x, y.
2x
2
+ 3y
2
− 11, x
2
− y
2
− 3 = x
2
− 4, y
2
− 1.
V (x + xy, y + xy, x
2

, y
2
) = V (x, y).
I(V (x
n
, y
m
)) = x, y n, m
I(V ) V ⊂ k
n
.
x
2
, y
2
 x
2
, y
2
 = I(V ) V
k
2
.
V = V (y −x
2
, z −x
3
) ⊂ k
3
.

V y
2
− xz ∈ I(V ).
y
2
− xz y − x
2
z −x
3
.
I(V (x −y)) = x − y.
V ⊂ R
3
(t, t
3
, t
4
), t ∈ R.
V
I(V ).
V ⊂ R
3
(t
2
, t
3
, t
4
), t ∈ R.
V

I(V ).
S ⊂ k
n
.
I(S) = {f ∈ k[x
1
, x
2
, . . . , x
n
] | f(a
1
, a
2
, . . . , a
n
) = 0 (a
1
, a
2
, . . . , a
n
) ∈ S}.
I(S)
I(S) S = {(a, a) ∈ R
2
| a = 1}.
I(Z
n
) Z

n
C
n
f ∈ k[x]
f(x) = a
0
x
m
+ a
1
x
m−1
+ ··· + a
m
,
a
i
∈ k a
0
= 0. a
0
x
m
f
LT(f) = a
0
x
m
.
f(x) = 3x

3
− 5x
2
+ 7. LT(f) = 3x
3
.
f, g deg f ≤ deg g
LT(g) LT(f).
k g ∈ k[x] f ∈ k[x]
q, r ∈ k[x]
f = qg + r,
r = 0 deg r < deg g.
k f ∈ k[x] f
deg f k.
k I ⊂ k[x].
f ∈ k[x] I = f. k[x]
f
1
, f
2
, . . . , f
s
∈ k[x],
GCD(f
1
, f
2
, . . . , f
s
), h

f
1
, f
2
, . . . , f
s
h.
f
1
, f
2
, . . . , f
s
p h p.
f
1
, f
2
, . . . , f
s
∈ k[x].
GCD(f
1
, f
2
, . . . , f
s
).
GCD(f
1

, f
2
, . . . , f
s
) f
1
, f
2
, . . . , f
s
.
GCD(f
1
, f
2
, . . . , f
s
).
x
3
− 3x + 2, x
4
− 1 x
6
− 1
x − 1.
x
3
− 3x + 2, x
4

− 1, x
6
− 1 = x − 1.
f
I = f
1
, f
2
, . . . , f
s
 h = GCD(f
1
, f
2
, . . . , f
s
).
f = qh + r, deg r < deg h. f ∈ I r = 0.
x
3
+4x
2
+3x−7 x
3
−3x+2, x
4
−1, x
6
−1 = x−1
x

3
+ 4x
2
+ 3x − 7 = (x
2
+ 5x + 8)(x − 1) + 1.
x, y k[x, y].
f, g ∈ k[x] h = GCD(f, g). A, B ∈ k[x]
Af + Bg = h.
f, g ∈ k[x]. f − qg, g = f, g q ∈ k[x].
f
1
, f
2
, . . . , f
s
∈ k[x] h = GCD(f
2
, . . . , f
s
). h = f
2
, . . . , f
s
,
f
1
, h = f
1
, f

2
, . . . , f
s
.
GCD(x
4
+ x
2
+ 1, x
4
− x
2
− 2x − 1, x
3
− 1.
GCD(x
3
+ 2x
2
− x − 2, x
3
− 2x
2
− x + 2, x
3
− x
2
− 4x + 4.
x
2

− 4 ∈ x
3
+ x
2
− 4x − 4, x
3
− x
2
− 4x + 4, x
3
− 2x
3
− x − 2?
V ⊂ C
f ∈ C[x] V (f) = ∅
f
f
1
, f
2
, . . . , f
s
∈ C[x]. V (f
1
, f
2
, . . . , f
s
) = ∅
GCD(f

1
, f
2
, . . . , f
s
) = 1.
f = c(x − a
1
)
r
1
(x − a
2
)
r
2
···(x − a
l
)
r
l
∈ C[x]
f
red
= c(x − a
1
)(x − a
2
) ···(x − a
l

).
V (f) = {a
1
, a
2
, . . . , a
l
}.
I(V (f)) = f
red
.
f = a
0
x
n
+ a
1
x
n−1
+ ··· + a
n
∈ C[x]
f

= na
0
x
n
+ (n − 1)a
1

x
n−2
+ ··· + a
n−1
+ 0.
(af)

= af

, c ∈ C,
(f + g)

= f

+ g

,
(fg)

= f

g + fg

.
f f

f ∈ C[x].
f = (x −a)
r
h C[x], h(a) = 0. f


= (x −a)
r−1
h
1
h
1
∈ C[x] a.
f = (x − a
1
)
r
1
(x − a
2
)
r
2
···(x − a
l
)
r
l
, a
i
f

= (x − a
1
)

r
1
−1
(x − a
2
)
r
2
−2
···(x − a
l
)
r
l
−1
H,
H ∈ C[x] a
i
GCD(f, f

) = (x − a
1
)
r
1
−1
(x − a
2
)
r

2
−2
···(x − a
l
)
r
l
−1
.
f
red
=
f
GCD(f, f

)
.
x
11
− x
10
+ 2x
8
− 4x
7
+ 3x
5
− 3x
4
+ x

3
+ 3x
2
− x − 1.
I(V (x
5
− 2x
4
+ 2x
2
− x, x
5
− x
4
− 2x
3
+ 2x
2
+ x − 1)).
k f ∈ k[x
1
, x
2
, . . . , x
n
]
k f g, h ∈ k[x
1
, x
2

, . . . , x
n
]
f = g ·h.
x
2
+ 1 Q, R C.
f ∈ k[x
1
, x
2
, . . . , x
n
]
k.
f ∈ k[x
1
, x
2
, . . . , x
n
] k g, h ∈
k[x
1
, x
2
, . . . , x
n
]. g · h f g f h
f.

f, g ∈ k[x
1
, x
2
, . . . , x
n
] deg
x
1
(f) > 0, deg
x
1
(g) > 0.
f, g h ∈ k[x
1
, x
2
, . . . , x
n
] deg
x
1
(h) > 0 f, g
k(x
2
, . . . , x
n
)[x
1
].

f ∈ k[x
1
, x
2
, . . . , x
n
].
f
1
, f
2
, . . . , f
r
k
f = f
1
· f
2
···f
r
.
f = g
1
· g
2
···g
s
g
i
r = s f

i
g
i
f, g ∈ k[x] l > 0 m > 0. f
g A, B ∈ k[x]
A, B
deg A ≤ m − 1 deg B ≤ l − 1.
Af + Bg = 0.
f, g
f = a
0
x
l
+ a
1
x
l−1
+ ··· + a
l
, a
0
= 0,
g = b
0
x
m
+ b
1
x
m−1

+ ··· + b
m
, b
0
= 0.
(l + m) ×(l + m) :
Syl(f, g, x) =

















a
0
b
0
a
1

a
0
b
1
b
0
a
2
a
1
b
2
b
1
a
0
b
0
a
1
b
1
a
l
b
m
a
l
b
m

a
l
b
m

















m a
i
l b
j
f, g x.
Res(f, g, x) = det Syl(f, g, x)
f g x.
f, g ∈ k[x] Res(f, g, x)
f g. f, g

k[x] Res(f, g, x) = 0.
f = 2x
2
+ 3x + 1,
g = 7x
2
+ x + 3
Q[x].
Res(f, g, x) =




2 0 7 0
3 2 1 7
1 3 3 1
0 1 0 3




= 153 = 0.
f g
f, g ∈ k[x]
A, B ∈ k[x]
Af + Bg = Res(f, g, x).
A B f g.
f ∈ k[x] k.
f ∈ k[x] f k
f k.

x
2
− 2 Q
R.
x
4
+ 1 Q
R.
≥ 4.
k
k[x] 1.
f =

i
a
i
x
i
1
g =

i
b
i
x
i
1
, a
i
, b

i
∈ k[x
2
, x
3
, . . . , x
n
].
u ∈ k[x
2
, x
3
, . . . , x
n
]. f u k[x
1
, x
2
, . . . , x
n
]
k[x
2
, x
3
, . . . , x
n
] a
i
u.

g ·h =

i
c
i
x
i
1
. c
i
a
i
b
i
.
f ∈ k[x
1
, x
2
, . . . , x
n
].
f h
1
h
2
. . . h
s
f
i h

i
f.
f
1
, f
2
, . . . , f
r
k
f = f
1
· f
2
···f
r
.
f = g
1
· g
2
···g
s
g
i
r = s f
i
g
i
f = x
5

− 3x
4
− 2x
3
+ 3x
2
+ 7x + 6,
g = x
4
+ x
2
+ 1.
f g. Q[x]?
f = f
a
1
1
· f
a
2
2
···f
a
r
r
∈ k[x], f
i
∈ k[x] f
i
f

j
, i = j. k Q
USCLN(f, f

) = f
a
1
−1
1
· f
a
2
−1
2
···f
a
r
−1
r
.
k ⊃ Q f

= 0).
f, g ∈ C[x] f g
C Res(f, g, x) = 0.
f = a
0
x
l
+ a

1
x
l−1
+ ··· + a
l
∈ k[x] a
0
= 0 l > 0.
Disc(f) =
(−1)
l(l−1)/2
a
0
Res(f, f

, x)
f. f h ∈ k[x]
f h
2
) Disc(f ) = 0.
f = 6x
4
− 23x
3
+ 32x
2
− 19x + 4 ∈ C[x]
f = ax
2
+ bx + c.

f = 2x
2
+ 3x + 1,
g = 7x
2
+ x + 3.
USCLN(f, g).
A, B ∈ k[x] Af + Bg = 1.
f, g ∈ Z[x] Res(f, g, x) ∈ Z.
f = xy −1,
g = x
2
+ y
2
− 4.
A, B Af + Bg = 1.
f g
l = deg(f) > 0 g = b
0
Res(f, g, x) = b
l
0
.
Res(f, g, x) f = a
0
deg(g) = m > 0.
f = a
0
g = b
0

Res(f, g, x) =

0 a
0
= 0 b
0
= 0,
1 a
0
= 0 b
0
= 0.
Res(f, g, x) = (−1)
deg(f ) deg(g)
Res(g, f, x).
f = a
0
x
l
+ a
1
x
l−1
+ ···+ a
l
g = b
0
x
m
+ b

1
x
m−1
+ ···+ b
m
k[x]. l ≥ m.
˜
f = f − (a
0
/b
0
)x
l−m
g. deg(
˜
f) ≤ l − 1. deg(
˜
f) = l − 1
Res(f, g, x) = (−1)
m
b
0
Res(
˜
f, g, x).
Res(f, g, x) = (−1)
m(l−deg(
˜
f))
b

l−deg(
˜
f)
0
Res(
˜
f, g, x).
f = qg + r k[x] deg(r) < deg(g).
Res(f, g, x) = (−1)
m(l−deg(r))
b
l−deg(r)
0
Res(r, g, x).
1.
R
⊂ R I ⊂ R f
1
, f
2
, . . . , f
s
∈ I
I = f
1
, f
2
, . . . , f
s

.
R :
I
1
⊂ I
2
⊂ ··· ⊂ I
m
⊂ ···
N ≥ 1
I
N
= I
N+1
= I
N+2
= ··· .
R
R
R I R.
R/I
R k(R) R. 0 ∈ S ⊂ R
B = {a/b ∈ k(R) | a ∈ R, b = 1 b S}.
B
R
1. R R[x]
k[x
1
, x
2

, . . . , x
n
]
I ⊂ k[x
1
, x
2
, . . . , x
n
]
V (I) = {(a
1
, a
2
, . . . , a
n
) ∈ k
n
| f(a
1
, a
2
, . . . , a
n
) = 0 f ∈ I}.
I V (I)
V (I) I = f
1
, f
2

, . . . , f
s

V (I) = V (f
1
, f
2
, . . . , f
s
).
f
1
, f
2
, . . . , f
s
 = g
1
, g
2
, . . . , g
r
 ⊂ k[x
1
, x
2
, . . . , x
n
].
V (f

1
, f
2
, . . . , f
s
) = V (g
1
, g
2
, . . . , g
r
).
f ∈ k[x
1
, x
2
, . . . , x
n
]. f ∈ x
1
, x
2
, . . . , x
n

x
1
, x
2
, . . . , x

n
, f = k[x
1
, x
2
, . . . , x
n
].
V
1
⊃ V
2
⊃ V
3
⊃ ··· .
N V
N
= V
N+1
= V
N+2
= ··· .
f
1
, f
2
, . . . ∈ k[x
1
, x
2

, . . . , x
n
]. I = f
1
, f
2
, . . .
N
I = f
1
, f
2
, . . . , f
N
.
f
1
, f
2
, . . . ∈ k[x
1
, x
2
, . . . , x
n
]. V (f
1
, f
2
, . . .)

f
1
= f
2
= ··· = 0.
N V (f
1
, f
2
, . . .) = V (f
1
, f
2
, . . . , f
N
).
V (I(V )) = V.
I = x
2
− y, x
2
+ y −4 ⊂ C[x, y] V = V (I).
I = x
2
− y, x
2
− 2.
V (I) = {(p,
√
2, 2)}.

g, g
1
, g
2
∈ k[x
1
, x
2
, . . . , x
n
] g = g
1
g
2
V (f, g) = V (f, g
1
) ∪
V (f, g
2
) f ∈ k[x
1
, x
2
, . . . , x
n
].
R
3
V (y −x
2

, xz −y
2
) = V (y −x
2
, xz −x
4
).