a0
a1a3a2a3a4a6a5 a7a3a8a3a7a10a9a6a11
a12a14a13 11{1
1,a15 f(x1; ; xn)a16a18a17a18a19 Ka20a18a21 na22a18a23a18a24a18a25a18a26a18a27,
a28a30a29,
a31a18a32a18a33a18a34a18a17a18a19 Ka20a18a21 na22a18a25a18a26a18a27 g(x1; ; xn) a35 h(x1; ; xn),a36
f(x1; ; xn) = g(x1; ; xn)h(x1; ; xn);
a37 g(x
1; ; xn)a35 h(x1; ; xn)a38a18a39a18a16a18a23a18a24a18a25a18a26a18a27,
a40a18a41,
a15 deg f = m,deg g = k,deg h = l,a42
g = gp + gp+1 + + gk; h = hq + hq+1 + hl;
a43a30a44 g
i; hja45a18a46a18a47
i; ja24a18a23a18a24a18a25a18a26a18a27,a48 gp; hqa16
a45a18a49
a44
a24a18a17a18a50a18a51a18a21a18a23a18a24a18a25a18a26a18a27,k + l = m,
a37
f = gphq +
mX
t=p+q+1
0
@X
i+j=t
gihj
1
A:
a52a18a53a18a54 p + q < m
a55 fa56a18a16a18a23a18a24a18a25a18a26a18a27, a57 p + q = k + l = ma58a18a59a18a60 p = k,q = l,a52a18a53 g = gk; h = hla39
a16a18a23a18a24a18a25a18a26a18a27,
2,a15 f(x; y) 2 K[x; y],a28a30a29, a31a18a32 f(x; x) = 0,a37 x y j f(x; y).
a40a18a41,
a15 f(x; y) =
nP
k=0
ak(x)yk,a37
f(x; y) = f(x; y) f(x; x) =
nX
k=0
ak(x)(yk xk)
= (y x)
nX
k=1
ak(x)(yk 1 + yk 2x + + yxk 2 + xk 1):
a52a18a53 x y j f(x; y).
3.
a61a18a62a18a63a18a64a18a65a18a64a18a27,
1
x1 a1
1
x1 a2
1
x1 an1
x2 a1
1
x2 a2
1
x2 an:,,,,,:,,,,,,,,,,:,,,,,,,,,,:,,,,,,,
1
xn a1
1
xn a2
1
xn an
:
a66,
a67a18a68a18a65a18a64a18a27a18a69
a47
Dn(x1; ; xn; a1; ; an),a37
Dn(x1; ; xn; a1; ; an) = G(x1; ; xn; a1; ; an)F(x
1; ; xn; a1; ; an);
a43a30a44 G
a35 F a39a18a16 x1; ; xn; a1; ; ana21a18a25a18a26a18a27, a70a18a71
F(x1; ; xn; a1; ; an) =
Y
16i;j6n
(xi aj):
a72a74a73
Dn(x1; ; xi; ; xi; ; xn; a1; ; an) = 0; Dn(x1; ; xn; a1; ; ai; ; ai; ; an) = 0;
1
a58a18a75
G(x1; ; xn; a1; ; an) =
Y
16i<j6n
(xi xj)
Y
16i<j6n
(aj ai) G1(x1; ; xn; a1; ; an):
a76a74a77a18a78a18a79 x
ia35 aja21a18a24a18a17 (a39a18a16 n 1a24 ),a58a18a71 G1 = cna16a18a80a18a81a18a82a18a17,
a52a18a53
Dn(x1; ; xn; a1; ; an) =
Q
16i<j6n
(xi xj) Q
16i<j6n
(aj ai) cn
Q
16i;j6n
(xi aj),
a83 a52
[(xn an)Dn(x1; ; xn; a1; ; an)]xn=an = Dn 1(x1; ; xn 1; a1; ; an 1);
a84a18a85
Q
16i<j6n 1
(xi xj)
!
(x1 an) (xn 1 an)
Q
16i<j6n 1
(aj ai)
!
(an a1) (an an 1) cn
Q
16i;j6n 1
(xi aj)
!
(an a1) (an an 1)(x1 an) (xn 1 an)
=
Q
16i<j6n 1
(xi xj) Q
16i<j6n 1
(aj ai) cn
Q
16i;j6n 1
(xi aj) = Dn 1(x1; ; xn 1; a1; ; an 1);
a58a18a75 cn = cn 1,a86
a53a18a87
a59,a50a18a88a18a58a18a75 cn = c1 = 1,a52 a57
Dn(x1; ; xn; a1; ; an) =
Q
16i<j6n
(xi xj)(aj ai)
Q
16i;j6n
(xi aj),
a12a14a13 11{2
1,a89a18a90a18a91a18a92a18a93a18a25a18a26a18a27a18a94a18a95a18a63a18a64a18a92a18a93a18a25a18a26a18a27,
(1) x21x2 + x1x22 + x21x3 + x1x23 + x22x3 + x2x23;
(2) x21x22 + x21x23 + x21x24 + x22x23 + x22x24 + x23x24;
(3) (x1 + x2)(x1 + x3)(x2 + x3);
(4) (x1x2 + x3)(x1x3 + x2)(x2x3 + x1);
(5) (x21 + x22)(x21 + x23)(x22 + x23);
(6) (x1 + x2 + x1x2)(x2 + x3 + x2x3)(x1 + x3 + x1x3).
a66, (1)
a68a96a27 = x1x2(x1 +x2 +x3)+x1x3(x1 +x2 +x3)+x2x3(x1 +x2 +x3) 3x1x2x3 = 1 2 3 3.
(2) 2 2 0 0 22
2 1 1 0 1 3
1 1 1 1 4
a52a18a53
a68a18a27 = 22 + A 1 3 + B 4.
a97 x
1 = x2 = x3 = 1,x4 = 0,a75 3 = 9 + 3A,A = 2;
a97 x
1 = x2 = x3 = x4 = 1,a75 B = 2;
a98
a68a18a27 = 22 2 1 3 + 2 4.
(3) a68a18a27 = ( 1 x3)( 1 x2)( 1 x1) = 31 1 21 + 2 1 3 = 1 2 3.
(4) a68a18a27 = 1
3
( 3 + x23)( 3 + x22)( 3 + x21),a72a74a73
x21 + x22 + x23 = 21 2 2;
2
x21x22 + x21x23 + x22x23 = 22 2 1 3;
x21x22x23 = 23;
a68a18a27 = 1
3
( 33 + ( 1 2 2) 23 + ( 22 1 3) 3 + 23)
= 21 3 2 1 3 + 22 2 2 3 + 23 + 3:
(5) a68a18a27 = (x21 + x22 + x23 x23)(x21 + x22 + x23 x22)(x21 + x22 + x23 x21)
= ( 21 2 2 x23)( 21 2 2 x22)( 21 2 2 x21)
= ( 21 2 2 x23)3 ( 21 2 2 x23)( 21 2 2 x23)2 + ( 21 2 1 3)( 21 2 2 x23) 23
= 21 22 2 31 3 2 32 + 4 1 2 3 23
(6) a68a18a27 = 1
3
[( 3 + 2)3 2( 3 + 2)2 + 1 3( 3 + 2) 23]
= 1
3
[ 3( 3 + 2)2 + 1 3( 3 + 2) 23]
= 22 + 2 2 3 + 23 + 1 3 + 1 2 23:
2,a89a18a90a18a91a18a92a18a93a18a25a18a26a18a27a18a94a18a95a18a63a18a64 na22a18a92a18a93a18a25a18a26a18a27,
(1)Px41; (2)Px21x22;
(3)Px21x2x3; (4)Px21x22x3x4:
a66, (1) 4
1 4 21 2 + 2 22 + 4 1 2 4 4.
(2) 22 2 1 3 + 2 4.
(3) 1 3 4 4.
(4) 2 4 4 1 5 + 9 6.
3,a15 x1; x2; x3a16a18a99a18a100 3x3 5x2 + 1a21a18a101a18a81a18a102, a61a18a62
x31x2 + x1x32 + x31x3 + x1x33 + x32x3 + x2x33:
a66,
a68a18a27 = 21 2 2 22 1 3 = 59,
4,a15 xyz 6= 0,a48 x + y + z = 0,a103,
x
y +
y
x +
x
z +
z
x +
y
z +
z
y:
a66,
a68a18a27 = 1xyz (x2z + y2z + x2y + yz2 + xy2 + xz2)
= 1xyz ((x + y + z)(xy + xz + yz) 3xyz)
= 3:
5,a28a30a29, a101a18a24a18a99a18a100 x3 + a1x2 + a2x + a3 = 0a21a18a101a18a81a18a102a18a104a18a91a18a105a18a17a18a64a18a21a18a106
a45a18a107a18a108a18a109a18a110
a16
2a31 9a1a2 + 27a3 = 0:
a40a18a41,
a101a18a81a18a102a18a104a18a91a18a105a18a17a18a64a18a21a18a106
a45a18a107a18a108a18a109a18a110
a16
a85
a63 3a81a18a17
x1 + x2 2x3; x1 + x3 2x2; x2 + x3 2x1;
a44a74a111a18a112a18a113
a80a18a81a18a91
a73 0,a98
a101a18a81a18a102a18a104a18a91a18a105a18a17a18a64 () (x1 + x2 2x3)(x1 + x3 2x2)(x2 + x3 2x1) = 0:
3
a57
(x1 + x2 2x3)(x1 + x3 2x2)(x2 + x3 2x1)
= (x1 + x2 + x3 3x3)(x1 + x2 + x3 3x2)(x1 + x2 + x3 3x1)
= ( a1)3 3( a1)( a1)2 + 9a2( a1) 27( a3)
= 2a31 9a1a2 + 27a3:
6,a114 na24a18a25a18a26a18a27 f(x)a21a18a102
a47
x1; x2; ; xn,a57a18a17 ca56a18a16 f(x)a21a18a102,a28a30a29,
nX
i=1
1
xi c =
f0(c)
f(c),
a40a18a41,
a115a18a116a18a25a18a26a18a27 f(x) = (x x1)(x x2) (x xn),a37
f0(x) =
nX
i=1
f(x)
x xi ;
f0(x)
f(x) =
nX
i=1
1
x xi ;
a117
a57
nX
i=1
1
xi c =
f0(c)
f(c),
7.
a15 x1; x2; ; xna16a18a99a18a100
xn + a1xn 1 + + an = 0
a21a18a102,a28a30a29, x2; x3; ; xna21a18a92a18a93a18a25a18a26a18a27a18a58a18a94a18a104 x1a35 a1; a2; ; ana21a18a25a18a26a18a27,
a40a18a41,
a15
f(x) = (x x1)(x x2) (x xn) =
nX
k=0
( 1)kakxn k:
a117
a57
(x x2) (x xn) = f(x)x x
1
= f(x) f(x1)x x
1
=
nP
k=0
( 1)kakxn k
nP
k=0
( 1)kakxn k1
x x1
=
n 1X
k=0
( 1)kak(xn k 1 + xn k 2x1 + + xn n 1):
a72
a50a18a118a18a80a18a27a18a71 xa21a18a119a18a24a18a26a18a120a18a17a18a39a18a16 x1a35 a1; ; ana21a18a25a18a26a18a27 (a0 = 1),a117 a57 x2; ; xna21a18a90a18a91a18a92a18a93a18a25a18a26
a27a18a16 x1a35 a1; ; ana21a18a25a18a26a18a27,a121a18a57
a72
a92a18a93a18a25a18a26a18a27a18a122a18a123a18a124a18a125a18a71 x2; ; xna21a18a92a18a93a18a25a18a26a18a27a18a58a18a94a18a104a18a16 x1a35
a1; ; ana21a18a25a18a26a18a27,
8.
a15
f(x) = (x x1)(x x2) (x xn)
= xn 1xn 1 + + ( 1)n n;
sk = xk1 + xk2 + + xkn; (k = 0; 1; 2; ):
(1) a28a30a29,
xk+1f0(x) = (s0xk + s1xk 1 + + sk 1x + sk)f(x) + g(x);
a43a30a44 g(x)
a21a18a24a18a17 < na126 g(x) = 0.
4
(2) a28a30a29a128a127a18a129 (Newton)a130a18a131,
sk 1sk 1 + 2sk 2 + + ( 1)k 1 k 1s1 + ( 1)kk k = 0 k 6 n;
sk 1sk 1 + + ( 1)n nsk n = 0 k > n:
a40a18a41,
a15 g(x) =
nP
i=1
xk+1i f(x)
x xi,
a37 g(x) = 0
a126 deg g(x) < n,a57
xk+1f0(x) g(x) =
nX
i=1
xk+1f(x)
x xi
nX
i=1
xk+1i f(x)
x xi =
xk+1 xk+1i
x xi
!
f(x)
=
nX
i=1
kX
j=0
(xk jxji f(x) =
kX
j=0
nX
i=1
(xk jxji
!
f(x)
=
0
@
kX
j=0
xk jsj
1
Af(x) = (s0xk + s1xk 1 + + sk 1x + sk)f(x):
a132
a75
a84 a28,
(2) a76a74a77 a91a18a27
xk+1f0(x) = (s0xk + s1xk 1 + + sk 1x + sk)f(x) + g(x)
a78a18a79 n
a24a18a26a18a120a18a17,a72a74a73 g(x)a21a18a24a18a17 < na126 g(x) = 0,a84a18a85
xk+1f0(x)a21 na24a18a26a18a120a18a17 = (s0xk + s1xk 1 + + sk 1x + sk)f(x)a21 na24a18a26a18a120a18a17,
a84a18a85 a54 k 6 n
a55,
(n k)( 1)k k = sk 1sk 1 + 2sk 2 + + ( 1)k ks0;
a132
sk 1sk 1 + 2sk 2 + + ( 1)k 1 k 1s1 + ( 1)kk k = 0:
a54 k > n,
0 = sk 1sk 1 + 2sk 2 + + ( 1)n nsk n;
a132
a75
a84 a28,
9.
a89a18a90a18a91a18a92a18a93a18a25a18a26a18a27a18a94a18a95 s2; s3; s4; s5.
a66, s
2 = 21 2 2,
s3 = 31 3 1 2 + 3 3,
s4 = 41 4 21 2 + 2 22 + 4 1 3 4 4,
s5 = 51 5 31 2 + 5 1 22 + 5 21 3 5 2 3 5 1 4 + 5 5.
a12a14a13 11{3
1,a28a30a29a74a133 a27a18a21a18a63a18a64a18a134a18a135, a15 f(x); g(x)
a45a18a46
a16 na24a18a35 ma24a18a25a18a26a18a27, a37
(1) Res(f; g) = ( 1)mn Res(g; f);
(2) Res(af; bg) = ambn Res(f; g);
(3) Res((x a)f; g) = g(a) Res(f; g).
a40a18a41, (1),(2)
a136a18a137, a138
a28 (3).
a15
f(x) = a0xn + a1xn 1 + + an; g(x) = b0xm + b1xm 1 + + bm;
5
a37
(x a)f(x) = a0xn+1 + (a1 a0a)xn + + (an an 1a)x ana:
Res((x a)f; g)=
a0 a1 a0a a2 a1a an an 1a ana
a0 a1 a0a a2 a1a an an 1a ana
...,..,..,..,..
9
>>>
>=
>>>
>;
n
a0 an an 1a ana
b0 b1 b2 bm 1 bm
b0 b1 b2 bm 1 bm
...,..,..,..,..
9
>>>
=
>>>;
m+1
b0 bm 1 bm
a139a141a140
a80a18a64a18a142,a119a18a64a18a143 aa144a18a145a18a118a18a80a18a64,a146
a111
a50a18a118a18a80a18a64,a58a18a75
Res((x a)f; g) =
a0 a1 a2 an 0
a0 a1 a2 an 0
...,..,..,..,..
9
>>=
>>>;
n
a0 an 0
b0 b1+b0a b2+b1a+b0a2 g(a) g(a)am
b0 b1+b0a g(a)am 1
...,..,..,..,..,..
9
>>>
=
>>>;
m+1
b0 g(a)
a117
a50a18a118a18a80a18a65a18a142,a119a18a65a18a143 ( a)a144a18a145a18a147a18a80a18a65,a146a18a145
a140 n + 1
a65,a148a18a149a18a50a18a118a18a80a18a64a18a150a18a151,a58a18a75
Res((x a)f; g) = g(a) Res(f; g):
2,a15 f(x) = a(x x1) (x xn),g(x) = b(x y1) (x ym).
a28a30a29, Res(f; g) = am
nQ
i=1
g(xi) = ( 1)mnbn
mQ
j=1
f(yj) = ambn
nQ
i=1
mQ
j=1
(xi yj).
a40a18a41, Res(f; g)= am Res((x x
1) (x xn); g(x))
= amg(x1) Res((x x2) (x xn); g(x))
= amg(x1)g(x2) g(xn)
= ambn
nQ
i=1
mQ
j=1
(xi yj) = ( 1)mnbn
mQ
j=1
f(yj):
3,a28a30a29, Res(f(x); g1(x)g2(x)) = Res(f(x); g1(x)) Res(f(x); g2(x)).
a40a18a41,
a15
g1(x) = a0(x a1) (x an); g2(x) = b0(x b1) (x bm);
a37
Res(f; g1g2) = ( 1)deg f(deg g1g2) Res(g1g2; f)
= ( 1)deg f(deg g1g2)adeg f0 bdeg f0
nY
i=1
f(ai)
mY
j=1
f(bj)
= ( 1)deg f(deg g1g2) Res(g1; f) Res(g2; f)
= Res(f; g1) Res(f; g2):
4,a15 f
a47a18a152
a80a18a25a18a26a18a27,a28a30a29, a92a18a153a18a154a18a25a18a26a18a27 h,Res(f; g) = Res(f; g + hf).
6
a40a18a41,
a15 f(x) = (x x1)(x x2) (x xn),m = deg g,a37
Res(f; g + hf) =
nY
i=1
(g(xi) + h(xi)f(xi))
=
nY
i=1
g(xi) = Res(f; g):
5,a15 f(x) = a0xn + a1xn 1 + + an 1x + an 2 K[x],
a28a30a29, f(x)
a21a18a155
a46
a27
D(f) = ( 1)n(n 1)2 a 10 Res(f; f0):
a40a18a41, D(f)= a2n 2
0
Q
16i<j6n
(xi xj)2 = ( 1)n(n 1)2 a2n 20 Q
i6=j
(xi xj)
= ( 1)n(n 1)2 a2n 20
nQ
i=1
(xi x1) (xi xi 1)(xi xi+1) (xi xn)
= ( 1)n(n 1)2 an 20
nQ
i=1
f0(xi)
= ( 1)n(n 1)2 an 20 Res((x x1) (x xn); f0)
= ( 1)n(n 1)2 a 10 Res(f; f0):
6,a61a18a62a18a63a18a64a18a25a18a26a18a27a18a21 a133 a27,
(1) f(x) = x3 3x2 + 2x + 1,g(x) = 2x2 x 1;
(2) f(x) = 2x3 3x2 x + 2,g(x) = x4 2x3 3x + 4;
(3) f(x) = xn + x + 1,g(x) = x2 3x + 2;
(4) f(x) = xn + 1,g(x) = (x 1)n;
(5) f(x) = xn 1x 1,g(x) = xm 1x 1 ;
(6) f(x) = a0xn + a1xn 1 + + an 1x + an,
g(x) = a0xn 1 + a1xn 2 + + an 2x + an 1.
a66, (1) Res(f; g) = ( 1)2 3 Res(2x2 x 1; f) = ( 1)6 23 f
12
f(1) = 7.
(2) f(x); g(x)a113a18a156a18a157 a102 1,a84a18a85 a133 a27 Res(f; g) = 0.
(3) Res(f; g) = ( 1)2n(1 + 1 + 1)(2n + 2 + 1) = 3(2n + 3).
(4) Res(f; g) = ( 1)n 2n.
(5) (a) a31 (m; n) = d > 1,a37 xn 1x 1 a35 xm 1x 1 a113a18a156a18a157 a102,a52a18a53 Res(f; g) = 0.
(b) a31 (m; n) = 1,a56a18a158a18a15 n > m,a37 n = mq + r,0 6 r < m,a136a18a137 (m; r) = 1,a37
xn 1
x 1 =
xmqxr 1
x 1 =
(xmq 1)xr + xr 1
x 1 ;
a117
a57
Res
xn 1
x 1 ;
xm 1
x 1
= ( 1)m 1)(n+r) Res
xr 1
x 1 ;
xm 1
x 1
:
a159a18a160 a28a30a29 (m 1)(n + r)
a80a18a124a18a16a18a161a18a17,
a31 m 1a16a18a161a18a17,a37 a133a18a162 a104a18a163, a164a18a15 m 1a16a18a165a18a17,a37 m
a47
a161a18a17,a117 a57 na16a18a165a18a17,ra38a18a16a18a165a18a17,a73 a16
n + ra16a18a161a18a17, a117 a57
Res
xn 1
x 1 ;
xm 1
x 1
= Res
xr 1
x 1 ;
xm 1
x 1
:
7
a148a18a89 ra166 m,a102a18a167a18a168a18a169a18a170a18a166a18a171a18a21a18a68a18a125,a72 (m; r) = 1a58a18a75
Res
xr 1
x 1 ;
xm 1
x 1
= = Res
xr0 1
x 1 ; 1
!
= 1:
a132 a54 (m; n) = 1
a55 Res
xn 1
x 1 ;
xm 1
x 1
= 1.
(6) a72a74a73 f(x) = xg(x) + an,a84a18a85
Res(f; g) = ( 1)n(n 1) Res(g; f) = ( 1)n(n 1) Res(g; an) = an 1n,
7,a54 a97a18a172a18a173 a55,a63a18a64a18a25a18a26a18a27 a113a18a156a18a157 a102,
(1) f(x) = x3 x + 2,g(x) = x2 + x + 2;
(2) f(x) = x3 + x2 9,g(x) = x3 + x 3.
a66, (1) Res(f; g) = 4( + 1)2( 3),a98 a54 = 1
a126 3a55
a113a18a156a18a157
a102,
(2) Res(f; g) = 9( 2 + 12)( 2 + 2),a98 a54 = 2p3ia126 p2ia55
a113a18a156a18a157
a102,
8,a103a18a63a18a64a30a174a74a175a18a21a18a146a18a176a18a177a18a178a18a99a18a100,
(1) x = t2 + t 1,y = 2t2 + t 1;
(2) x = t 1t2 + 1,y = t2 + t 1t2 + 1,
a66, (1) 4x2 4xy + y2 + 5x 3y + 1 = 0.
(2) 5x2 6xy + 2y2 + 5x 3y + 1 = 0.
9,a54
a47
a172a18a173
a55,a63a18a64a18a25a18a26a18a27
a113a18a179
a102?
(1) f(x) = x3 3x + ; (2) f(x) = x4 4x3 + (2 )x2 + 2x 2.
a66, (1) 2; 2;
(2) 1; 32 ; 72 + 92 p3i; 72 92 p3i.
10,a103a18a63a18a64a18a99a18a100a18a180a18a21
a49
:
(1)
( 5x2 6xy + 5y2 = 16;
2x2 xy + y2 x y = 4; (2)
( x2 + y2 + 4x 2y = 3;
x2 + 4xy y2 + 10y = 9:
a66, (1) Res
y(f; g) = 32(y4 y3 3y2 + y + 2),(
x = 1
y = 1
( x = 1
y = 1
( x = 2
y = 2
(2) Resx(f; g) = 4(5x4 + 40x3 + 106x2 + 104x + 33),(
x = 1
y = 2
( x = 3
y = 0
( x = 2 + 3
5
p5
y = 1 + 15 p5
( x = 2 3
5
p5
y = 1 15 p5
11,a103a18a63a18a64a30a181a74a182a30a174a74a175a18a21a18a183a18a184a18a177a18a178,
(1) a181 x2 + y2 3x y = 0 a35a18a185a30a174a74a175 x2 + 2xy y2 4y 2 = 0;
(2) a185a30a174a74a175 4x2 7xy + y2 + 13x 2y 3 = 0 a35a18a185a30a174a74a175 9x2 14xy + y2 + 28x 4y 5 = 0.
a66, (1) (1; 1);
3
2 +
1
2
p2; 1
2 +
p2 ; 3
2
1
2
p2; 1
2
p2 ;
(2) (0; 1); (1; 2); (2; 3); ( 2; 1).
8
a1a3a2a3a4a6a5 a7a3a8a3a7a10a9a6a11
a12a14a13 11{1
1,a15 f(x1; ; xn)a16a18a17a18a19 Ka20a18a21 na22a18a23a18a24a18a25a18a26a18a27,
a28a30a29,
a31a18a32a18a33a18a34a18a17a18a19 Ka20a18a21 na22a18a25a18a26a18a27 g(x1; ; xn) a35 h(x1; ; xn),a36
f(x1; ; xn) = g(x1; ; xn)h(x1; ; xn);
a37 g(x
1; ; xn)a35 h(x1; ; xn)a38a18a39a18a16a18a23a18a24a18a25a18a26a18a27,
a40a18a41,
a15 deg f = m,deg g = k,deg h = l,a42
g = gp + gp+1 + + gk; h = hq + hq+1 + hl;
a43a30a44 g
i; hja45a18a46a18a47
i; ja24a18a23a18a24a18a25a18a26a18a27,a48 gp; hqa16
a45a18a49
a44
a24a18a17a18a50a18a51a18a21a18a23a18a24a18a25a18a26a18a27,k + l = m,
a37
f = gphq +
mX
t=p+q+1
0
@X
i+j=t
gihj
1
A:
a52a18a53a18a54 p + q < m
a55 fa56a18a16a18a23a18a24a18a25a18a26a18a27, a57 p + q = k + l = ma58a18a59a18a60 p = k,q = l,a52a18a53 g = gk; h = hla39
a16a18a23a18a24a18a25a18a26a18a27,
2,a15 f(x; y) 2 K[x; y],a28a30a29, a31a18a32 f(x; x) = 0,a37 x y j f(x; y).
a40a18a41,
a15 f(x; y) =
nP
k=0
ak(x)yk,a37
f(x; y) = f(x; y) f(x; x) =
nX
k=0
ak(x)(yk xk)
= (y x)
nX
k=1
ak(x)(yk 1 + yk 2x + + yxk 2 + xk 1):
a52a18a53 x y j f(x; y).
3.
a61a18a62a18a63a18a64a18a65a18a64a18a27,
1
x1 a1
1
x1 a2
1
x1 an1
x2 a1
1
x2 a2
1
x2 an:,,,,,:,,,,,,,,,,:,,,,,,,,,,:,,,,,,,
1
xn a1
1
xn a2
1
xn an
:
a66,
a67a18a68a18a65a18a64a18a27a18a69
a47
Dn(x1; ; xn; a1; ; an),a37
Dn(x1; ; xn; a1; ; an) = G(x1; ; xn; a1; ; an)F(x
1; ; xn; a1; ; an);
a43a30a44 G
a35 F a39a18a16 x1; ; xn; a1; ; ana21a18a25a18a26a18a27, a70a18a71
F(x1; ; xn; a1; ; an) =
Y
16i;j6n
(xi aj):
a72a74a73
Dn(x1; ; xi; ; xi; ; xn; a1; ; an) = 0; Dn(x1; ; xn; a1; ; ai; ; ai; ; an) = 0;
1
a58a18a75
G(x1; ; xn; a1; ; an) =
Y
16i<j6n
(xi xj)
Y
16i<j6n
(aj ai) G1(x1; ; xn; a1; ; an):
a76a74a77a18a78a18a79 x
ia35 aja21a18a24a18a17 (a39a18a16 n 1a24 ),a58a18a71 G1 = cna16a18a80a18a81a18a82a18a17,
a52a18a53
Dn(x1; ; xn; a1; ; an) =
Q
16i<j6n
(xi xj) Q
16i<j6n
(aj ai) cn
Q
16i;j6n
(xi aj),
a83 a52
[(xn an)Dn(x1; ; xn; a1; ; an)]xn=an = Dn 1(x1; ; xn 1; a1; ; an 1);
a84a18a85
Q
16i<j6n 1
(xi xj)
!
(x1 an) (xn 1 an)
Q
16i<j6n 1
(aj ai)
!
(an a1) (an an 1) cn
Q
16i;j6n 1
(xi aj)
!
(an a1) (an an 1)(x1 an) (xn 1 an)
=
Q
16i<j6n 1
(xi xj) Q
16i<j6n 1
(aj ai) cn
Q
16i;j6n 1
(xi aj) = Dn 1(x1; ; xn 1; a1; ; an 1);
a58a18a75 cn = cn 1,a86
a53a18a87
a59,a50a18a88a18a58a18a75 cn = c1 = 1,a52 a57
Dn(x1; ; xn; a1; ; an) =
Q
16i<j6n
(xi xj)(aj ai)
Q
16i;j6n
(xi aj),
a12a14a13 11{2
1,a89a18a90a18a91a18a92a18a93a18a25a18a26a18a27a18a94a18a95a18a63a18a64a18a92a18a93a18a25a18a26a18a27,
(1) x21x2 + x1x22 + x21x3 + x1x23 + x22x3 + x2x23;
(2) x21x22 + x21x23 + x21x24 + x22x23 + x22x24 + x23x24;
(3) (x1 + x2)(x1 + x3)(x2 + x3);
(4) (x1x2 + x3)(x1x3 + x2)(x2x3 + x1);
(5) (x21 + x22)(x21 + x23)(x22 + x23);
(6) (x1 + x2 + x1x2)(x2 + x3 + x2x3)(x1 + x3 + x1x3).
a66, (1)
a68a96a27 = x1x2(x1 +x2 +x3)+x1x3(x1 +x2 +x3)+x2x3(x1 +x2 +x3) 3x1x2x3 = 1 2 3 3.
(2) 2 2 0 0 22
2 1 1 0 1 3
1 1 1 1 4
a52a18a53
a68a18a27 = 22 + A 1 3 + B 4.
a97 x
1 = x2 = x3 = 1,x4 = 0,a75 3 = 9 + 3A,A = 2;
a97 x
1 = x2 = x3 = x4 = 1,a75 B = 2;
a98
a68a18a27 = 22 2 1 3 + 2 4.
(3) a68a18a27 = ( 1 x3)( 1 x2)( 1 x1) = 31 1 21 + 2 1 3 = 1 2 3.
(4) a68a18a27 = 1
3
( 3 + x23)( 3 + x22)( 3 + x21),a72a74a73
x21 + x22 + x23 = 21 2 2;
2
x21x22 + x21x23 + x22x23 = 22 2 1 3;
x21x22x23 = 23;
a68a18a27 = 1
3
( 33 + ( 1 2 2) 23 + ( 22 1 3) 3 + 23)
= 21 3 2 1 3 + 22 2 2 3 + 23 + 3:
(5) a68a18a27 = (x21 + x22 + x23 x23)(x21 + x22 + x23 x22)(x21 + x22 + x23 x21)
= ( 21 2 2 x23)( 21 2 2 x22)( 21 2 2 x21)
= ( 21 2 2 x23)3 ( 21 2 2 x23)( 21 2 2 x23)2 + ( 21 2 1 3)( 21 2 2 x23) 23
= 21 22 2 31 3 2 32 + 4 1 2 3 23
(6) a68a18a27 = 1
3
[( 3 + 2)3 2( 3 + 2)2 + 1 3( 3 + 2) 23]
= 1
3
[ 3( 3 + 2)2 + 1 3( 3 + 2) 23]
= 22 + 2 2 3 + 23 + 1 3 + 1 2 23:
2,a89a18a90a18a91a18a92a18a93a18a25a18a26a18a27a18a94a18a95a18a63a18a64 na22a18a92a18a93a18a25a18a26a18a27,
(1)Px41; (2)Px21x22;
(3)Px21x2x3; (4)Px21x22x3x4:
a66, (1) 4
1 4 21 2 + 2 22 + 4 1 2 4 4.
(2) 22 2 1 3 + 2 4.
(3) 1 3 4 4.
(4) 2 4 4 1 5 + 9 6.
3,a15 x1; x2; x3a16a18a99a18a100 3x3 5x2 + 1a21a18a101a18a81a18a102, a61a18a62
x31x2 + x1x32 + x31x3 + x1x33 + x32x3 + x2x33:
a66,
a68a18a27 = 21 2 2 22 1 3 = 59,
4,a15 xyz 6= 0,a48 x + y + z = 0,a103,
x
y +
y
x +
x
z +
z
x +
y
z +
z
y:
a66,
a68a18a27 = 1xyz (x2z + y2z + x2y + yz2 + xy2 + xz2)
= 1xyz ((x + y + z)(xy + xz + yz) 3xyz)
= 3:
5,a28a30a29, a101a18a24a18a99a18a100 x3 + a1x2 + a2x + a3 = 0a21a18a101a18a81a18a102a18a104a18a91a18a105a18a17a18a64a18a21a18a106
a45a18a107a18a108a18a109a18a110
a16
2a31 9a1a2 + 27a3 = 0:
a40a18a41,
a101a18a81a18a102a18a104a18a91a18a105a18a17a18a64a18a21a18a106
a45a18a107a18a108a18a109a18a110
a16
a85
a63 3a81a18a17
x1 + x2 2x3; x1 + x3 2x2; x2 + x3 2x1;
a44a74a111a18a112a18a113
a80a18a81a18a91
a73 0,a98
a101a18a81a18a102a18a104a18a91a18a105a18a17a18a64 () (x1 + x2 2x3)(x1 + x3 2x2)(x2 + x3 2x1) = 0:
3
a57
(x1 + x2 2x3)(x1 + x3 2x2)(x2 + x3 2x1)
= (x1 + x2 + x3 3x3)(x1 + x2 + x3 3x2)(x1 + x2 + x3 3x1)
= ( a1)3 3( a1)( a1)2 + 9a2( a1) 27( a3)
= 2a31 9a1a2 + 27a3:
6,a114 na24a18a25a18a26a18a27 f(x)a21a18a102
a47
x1; x2; ; xn,a57a18a17 ca56a18a16 f(x)a21a18a102,a28a30a29,
nX
i=1
1
xi c =
f0(c)
f(c),
a40a18a41,
a115a18a116a18a25a18a26a18a27 f(x) = (x x1)(x x2) (x xn),a37
f0(x) =
nX
i=1
f(x)
x xi ;
f0(x)
f(x) =
nX
i=1
1
x xi ;
a117
a57
nX
i=1
1
xi c =
f0(c)
f(c),
7.
a15 x1; x2; ; xna16a18a99a18a100
xn + a1xn 1 + + an = 0
a21a18a102,a28a30a29, x2; x3; ; xna21a18a92a18a93a18a25a18a26a18a27a18a58a18a94a18a104 x1a35 a1; a2; ; ana21a18a25a18a26a18a27,
a40a18a41,
a15
f(x) = (x x1)(x x2) (x xn) =
nX
k=0
( 1)kakxn k:
a117
a57
(x x2) (x xn) = f(x)x x
1
= f(x) f(x1)x x
1
=
nP
k=0
( 1)kakxn k
nP
k=0
( 1)kakxn k1
x x1
=
n 1X
k=0
( 1)kak(xn k 1 + xn k 2x1 + + xn n 1):
a72
a50a18a118a18a80a18a27a18a71 xa21a18a119a18a24a18a26a18a120a18a17a18a39a18a16 x1a35 a1; ; ana21a18a25a18a26a18a27 (a0 = 1),a117 a57 x2; ; xna21a18a90a18a91a18a92a18a93a18a25a18a26
a27a18a16 x1a35 a1; ; ana21a18a25a18a26a18a27,a121a18a57
a72
a92a18a93a18a25a18a26a18a27a18a122a18a123a18a124a18a125a18a71 x2; ; xna21a18a92a18a93a18a25a18a26a18a27a18a58a18a94a18a104a18a16 x1a35
a1; ; ana21a18a25a18a26a18a27,
8.
a15
f(x) = (x x1)(x x2) (x xn)
= xn 1xn 1 + + ( 1)n n;
sk = xk1 + xk2 + + xkn; (k = 0; 1; 2; ):
(1) a28a30a29,
xk+1f0(x) = (s0xk + s1xk 1 + + sk 1x + sk)f(x) + g(x);
a43a30a44 g(x)
a21a18a24a18a17 < na126 g(x) = 0.
4
(2) a28a30a29a128a127a18a129 (Newton)a130a18a131,
sk 1sk 1 + 2sk 2 + + ( 1)k 1 k 1s1 + ( 1)kk k = 0 k 6 n;
sk 1sk 1 + + ( 1)n nsk n = 0 k > n:
a40a18a41,
a15 g(x) =
nP
i=1
xk+1i f(x)
x xi,
a37 g(x) = 0
a126 deg g(x) < n,a57
xk+1f0(x) g(x) =
nX
i=1
xk+1f(x)
x xi
nX
i=1
xk+1i f(x)
x xi =
xk+1 xk+1i
x xi
!
f(x)
=
nX
i=1
kX
j=0
(xk jxji f(x) =
kX
j=0
nX
i=1
(xk jxji
!
f(x)
=
0
@
kX
j=0
xk jsj
1
Af(x) = (s0xk + s1xk 1 + + sk 1x + sk)f(x):
a132
a75
a84 a28,
(2) a76a74a77 a91a18a27
xk+1f0(x) = (s0xk + s1xk 1 + + sk 1x + sk)f(x) + g(x)
a78a18a79 n
a24a18a26a18a120a18a17,a72a74a73 g(x)a21a18a24a18a17 < na126 g(x) = 0,a84a18a85
xk+1f0(x)a21 na24a18a26a18a120a18a17 = (s0xk + s1xk 1 + + sk 1x + sk)f(x)a21 na24a18a26a18a120a18a17,
a84a18a85 a54 k 6 n
a55,
(n k)( 1)k k = sk 1sk 1 + 2sk 2 + + ( 1)k ks0;
a132
sk 1sk 1 + 2sk 2 + + ( 1)k 1 k 1s1 + ( 1)kk k = 0:
a54 k > n,
0 = sk 1sk 1 + 2sk 2 + + ( 1)n nsk n;
a132
a75
a84 a28,
9.
a89a18a90a18a91a18a92a18a93a18a25a18a26a18a27a18a94a18a95 s2; s3; s4; s5.
a66, s
2 = 21 2 2,
s3 = 31 3 1 2 + 3 3,
s4 = 41 4 21 2 + 2 22 + 4 1 3 4 4,
s5 = 51 5 31 2 + 5 1 22 + 5 21 3 5 2 3 5 1 4 + 5 5.
a12a14a13 11{3
1,a28a30a29a74a133 a27a18a21a18a63a18a64a18a134a18a135, a15 f(x); g(x)
a45a18a46
a16 na24a18a35 ma24a18a25a18a26a18a27, a37
(1) Res(f; g) = ( 1)mn Res(g; f);
(2) Res(af; bg) = ambn Res(f; g);
(3) Res((x a)f; g) = g(a) Res(f; g).
a40a18a41, (1),(2)
a136a18a137, a138
a28 (3).
a15
f(x) = a0xn + a1xn 1 + + an; g(x) = b0xm + b1xm 1 + + bm;
5
a37
(x a)f(x) = a0xn+1 + (a1 a0a)xn + + (an an 1a)x ana:
Res((x a)f; g)=
a0 a1 a0a a2 a1a an an 1a ana
a0 a1 a0a a2 a1a an an 1a ana
...,..,..,..,..
9
>>>
>=
>>>
>;
n
a0 an an 1a ana
b0 b1 b2 bm 1 bm
b0 b1 b2 bm 1 bm
...,..,..,..,..
9
>>>
=
>>>;
m+1
b0 bm 1 bm
a139a141a140
a80a18a64a18a142,a119a18a64a18a143 aa144a18a145a18a118a18a80a18a64,a146
a111
a50a18a118a18a80a18a64,a58a18a75
Res((x a)f; g) =
a0 a1 a2 an 0
a0 a1 a2 an 0
...,..,..,..,..
9
>>=
>>>;
n
a0 an 0
b0 b1+b0a b2+b1a+b0a2 g(a) g(a)am
b0 b1+b0a g(a)am 1
...,..,..,..,..,..
9
>>>
=
>>>;
m+1
b0 g(a)
a117
a50a18a118a18a80a18a65a18a142,a119a18a65a18a143 ( a)a144a18a145a18a147a18a80a18a65,a146a18a145
a140 n + 1
a65,a148a18a149a18a50a18a118a18a80a18a64a18a150a18a151,a58a18a75
Res((x a)f; g) = g(a) Res(f; g):
2,a15 f(x) = a(x x1) (x xn),g(x) = b(x y1) (x ym).
a28a30a29, Res(f; g) = am
nQ
i=1
g(xi) = ( 1)mnbn
mQ
j=1
f(yj) = ambn
nQ
i=1
mQ
j=1
(xi yj).
a40a18a41, Res(f; g)= am Res((x x
1) (x xn); g(x))
= amg(x1) Res((x x2) (x xn); g(x))
= amg(x1)g(x2) g(xn)
= ambn
nQ
i=1
mQ
j=1
(xi yj) = ( 1)mnbn
mQ
j=1
f(yj):
3,a28a30a29, Res(f(x); g1(x)g2(x)) = Res(f(x); g1(x)) Res(f(x); g2(x)).
a40a18a41,
a15
g1(x) = a0(x a1) (x an); g2(x) = b0(x b1) (x bm);
a37
Res(f; g1g2) = ( 1)deg f(deg g1g2) Res(g1g2; f)
= ( 1)deg f(deg g1g2)adeg f0 bdeg f0
nY
i=1
f(ai)
mY
j=1
f(bj)
= ( 1)deg f(deg g1g2) Res(g1; f) Res(g2; f)
= Res(f; g1) Res(f; g2):
4,a15 f
a47a18a152
a80a18a25a18a26a18a27,a28a30a29, a92a18a153a18a154a18a25a18a26a18a27 h,Res(f; g) = Res(f; g + hf).
6
a40a18a41,
a15 f(x) = (x x1)(x x2) (x xn),m = deg g,a37
Res(f; g + hf) =
nY
i=1
(g(xi) + h(xi)f(xi))
=
nY
i=1
g(xi) = Res(f; g):
5,a15 f(x) = a0xn + a1xn 1 + + an 1x + an 2 K[x],
a28a30a29, f(x)
a21a18a155
a46
a27
D(f) = ( 1)n(n 1)2 a 10 Res(f; f0):
a40a18a41, D(f)= a2n 2
0
Q
16i<j6n
(xi xj)2 = ( 1)n(n 1)2 a2n 20 Q
i6=j
(xi xj)
= ( 1)n(n 1)2 a2n 20
nQ
i=1
(xi x1) (xi xi 1)(xi xi+1) (xi xn)
= ( 1)n(n 1)2 an 20
nQ
i=1
f0(xi)
= ( 1)n(n 1)2 an 20 Res((x x1) (x xn); f0)
= ( 1)n(n 1)2 a 10 Res(f; f0):
6,a61a18a62a18a63a18a64a18a25a18a26a18a27a18a21 a133 a27,
(1) f(x) = x3 3x2 + 2x + 1,g(x) = 2x2 x 1;
(2) f(x) = 2x3 3x2 x + 2,g(x) = x4 2x3 3x + 4;
(3) f(x) = xn + x + 1,g(x) = x2 3x + 2;
(4) f(x) = xn + 1,g(x) = (x 1)n;
(5) f(x) = xn 1x 1,g(x) = xm 1x 1 ;
(6) f(x) = a0xn + a1xn 1 + + an 1x + an,
g(x) = a0xn 1 + a1xn 2 + + an 2x + an 1.
a66, (1) Res(f; g) = ( 1)2 3 Res(2x2 x 1; f) = ( 1)6 23 f
12
f(1) = 7.
(2) f(x); g(x)a113a18a156a18a157 a102 1,a84a18a85 a133 a27 Res(f; g) = 0.
(3) Res(f; g) = ( 1)2n(1 + 1 + 1)(2n + 2 + 1) = 3(2n + 3).
(4) Res(f; g) = ( 1)n 2n.
(5) (a) a31 (m; n) = d > 1,a37 xn 1x 1 a35 xm 1x 1 a113a18a156a18a157 a102,a52a18a53 Res(f; g) = 0.
(b) a31 (m; n) = 1,a56a18a158a18a15 n > m,a37 n = mq + r,0 6 r < m,a136a18a137 (m; r) = 1,a37
xn 1
x 1 =
xmqxr 1
x 1 =
(xmq 1)xr + xr 1
x 1 ;
a117
a57
Res
xn 1
x 1 ;
xm 1
x 1
= ( 1)m 1)(n+r) Res
xr 1
x 1 ;
xm 1
x 1
:
a159a18a160 a28a30a29 (m 1)(n + r)
a80a18a124a18a16a18a161a18a17,
a31 m 1a16a18a161a18a17,a37 a133a18a162 a104a18a163, a164a18a15 m 1a16a18a165a18a17,a37 m
a47
a161a18a17,a117 a57 na16a18a165a18a17,ra38a18a16a18a165a18a17,a73 a16
n + ra16a18a161a18a17, a117 a57
Res
xn 1
x 1 ;
xm 1
x 1
= Res
xr 1
x 1 ;
xm 1
x 1
:
7
a148a18a89 ra166 m,a102a18a167a18a168a18a169a18a170a18a166a18a171a18a21a18a68a18a125,a72 (m; r) = 1a58a18a75
Res
xr 1
x 1 ;
xm 1
x 1
= = Res
xr0 1
x 1 ; 1
!
= 1:
a132 a54 (m; n) = 1
a55 Res
xn 1
x 1 ;
xm 1
x 1
= 1.
(6) a72a74a73 f(x) = xg(x) + an,a84a18a85
Res(f; g) = ( 1)n(n 1) Res(g; f) = ( 1)n(n 1) Res(g; an) = an 1n,
7,a54 a97a18a172a18a173 a55,a63a18a64a18a25a18a26a18a27 a113a18a156a18a157 a102,
(1) f(x) = x3 x + 2,g(x) = x2 + x + 2;
(2) f(x) = x3 + x2 9,g(x) = x3 + x 3.
a66, (1) Res(f; g) = 4( + 1)2( 3),a98 a54 = 1
a126 3a55
a113a18a156a18a157
a102,
(2) Res(f; g) = 9( 2 + 12)( 2 + 2),a98 a54 = 2p3ia126 p2ia55
a113a18a156a18a157
a102,
8,a103a18a63a18a64a30a174a74a175a18a21a18a146a18a176a18a177a18a178a18a99a18a100,
(1) x = t2 + t 1,y = 2t2 + t 1;
(2) x = t 1t2 + 1,y = t2 + t 1t2 + 1,
a66, (1) 4x2 4xy + y2 + 5x 3y + 1 = 0.
(2) 5x2 6xy + 2y2 + 5x 3y + 1 = 0.
9,a54
a47
a172a18a173
a55,a63a18a64a18a25a18a26a18a27
a113a18a179
a102?
(1) f(x) = x3 3x + ; (2) f(x) = x4 4x3 + (2 )x2 + 2x 2.
a66, (1) 2; 2;
(2) 1; 32 ; 72 + 92 p3i; 72 92 p3i.
10,a103a18a63a18a64a18a99a18a100a18a180a18a21
a49
:
(1)
( 5x2 6xy + 5y2 = 16;
2x2 xy + y2 x y = 4; (2)
( x2 + y2 + 4x 2y = 3;
x2 + 4xy y2 + 10y = 9:
a66, (1) Res
y(f; g) = 32(y4 y3 3y2 + y + 2),(
x = 1
y = 1
( x = 1
y = 1
( x = 2
y = 2
(2) Resx(f; g) = 4(5x4 + 40x3 + 106x2 + 104x + 33),(
x = 1
y = 2
( x = 3
y = 0
( x = 2 + 3
5
p5
y = 1 + 15 p5
( x = 2 3
5
p5
y = 1 15 p5
11,a103a18a63a18a64a30a181a74a182a30a174a74a175a18a21a18a183a18a184a18a177a18a178,
(1) a181 x2 + y2 3x y = 0 a35a18a185a30a174a74a175 x2 + 2xy y2 4y 2 = 0;
(2) a185a30a174a74a175 4x2 7xy + y2 + 13x 2y 3 = 0 a35a18a185a30a174a74a175 9x2 14xy + y2 + 28x 4y 5 = 0.
a66, (1) (1; 1);
3
2 +
1
2
p2; 1
2 +
p2 ; 3
2
1
2
p2; 1
2
p2 ;
(2) (0; 1); (1; 2); (2; 3); ( 2; 1).
8
