Phương pháp xác định phương trình dạng ẩn cho đường cong và mặt tham số hữu tỷ4_2

Chuang 2

L Y THUYET KHU, SYZYGYvA BAI TOAN AN HOA
Trong chuang nay, chung t6i giai thi~u co sd ly thuye't cua mQt s6 phuong
phap tie'p c~n bai toan fin h6a duong cong va m<)ttham s6 hUll ty. NQi dung cua
Chuang 2 g6m c6 2 ml;1c:

. Ml;1c2.1 trlnh bay ly thuye't khli va ling dl;1ngvao bai toan fin h6a.
. Ml;1c2.2 trlnh bay m6i lien h~ giua cac syzygy! va phuong trlnh fin cua cac
duong cong va m<)t tham s6.
2.1 Ly thuye't khd' va bai toaD Gn hoa
Ly thuye't khli' nghien CUllcac ky thu~t va cac giai thu~t khli' bie'n cua mQt
h~ cac phuong trlnh da thuc. Ly thuye't khli c6 th~ du<;Jcling dl;1ngd~ giai quye't
nhi€u bai toan thie't thl;(ctrong nhi€u IInh vl;(c,bai toan fin h6a Ia mQt ling dl;1ng
cl;1th~. v€ co ban c6 ba huang tie'p c~n chlnh d6i vai vi~c khli bie'n cua mQt h~
cac phuong trlnh da thuc. Thu nhfft, phuong phap ke't thuc dl;(atren ly thuye't v€
cac dinh thuc du<;Jcnghien CUllbdi Macaulay, Bezout, Dixon va nhi€u nha toan
hQc khac. Thu hai, phuong phap co sd Grabner dl;(a tren ly thuye't ideal da thuc
du<;Jcd€ xufft bdi Buchberger.

Thu ba, phuong phap dl;(a tren t~p d<)c trung cua

Ritt du<;Jcd€ xufft bdi Wu. Trang ml;1cnay chung Wi se trlnh bay mQt cach ng3:n

gQn hai phuong pha p khli' bie'n dl;(atren ke't thuc va co sd Grobner va cac ke't
qua chinh cua ly thuye't khli'ling dl;1ngtrang bai toan fin h6a.

1 Syzygy c6 ngu6n g6c ti'r titng Hy Ll).p l!i mQt thu~t ngu thien van hQc di n6i dtn str ktt h<;ipcua cac

hanh tinh. Trang roan hQc syzygy mo tel m6i quail ht$ tuytn tinh giua cac phffn tii' sinh cua mQt modun.

10

2.1.1 Ke't thU'c va bai toaD iin hoa
D~u tien, chung ta xet tru'ong hQp ke't thuc cua hai da thuc mQt bie'n va
cac tinh cha't cd ban cua no.

Blnh nghia 2.1. Cho hai da thac I, g E k[x]

1 =amx

m

+..,+a1x+aO'

g = bnxn

V(ji am, bn +=0 va

+ ,.. + b1x+ bo

m, n

> O. Khi do kit thac cua I va g, du<;lcky hifu fa

Res(f, g), fa dtnh thac clip (m + n) x (m + n) sau
am

bn

am-1

bn-1

am
am-1

bn-1

'.

Res(f, g) = lao

bn

ao

bo

am
am-1

"

bo

-y
n cQt

bn-1

bo

ao
\.,.

bn

,..I\.,.

-y
m cQt

,..I

trong do cac phdn trffng fa cac sff khong.
M~nh d~ 2.2. Cho hai da thac I, g E k[x] co bqc duang,khi do
(1)

Res(f, g) fa m(}t da thac hf sff nguyen theo cac hf so' cua

I va g.

(2) I va g co nghifm chung khi va chi khi Res(f, g) = O.
(3)

Tbn t(}i hai da thac hf sff nguyen A, B E k[x] theo cac hf sff cua I va g saG

cho AI + Bg = Res(f, g).
Chung minh: xem [9, tr. 151

- 152].


11

NSu thugn nha't hai da thue f, g trd thanh F, G thi ta vftn co th€ dinh
nghia kSt thue eua F va G la Res(F, G) = Res(f, g) va Res(F, G) vftn thoa cae
Hnh eha't trong M~nh dS 2.2. Trong tru'ong hQp t6ng quat ta co th€ xfty dl,l'ngkSt
thue eho cae da thue nhiSu biSn nhu' sau.
Dinh nghia 2.3. Cha n + 1 da thac Fo,...,F;! E k[xo,...,xnJ thudn nhttt bgc dO,...,dn

tl1dngang. Khi do cac F; co thi vift dl1did{Jngsau
F;

I:

=

Ci,aXa,

lal=d,

trang do a = (ao,...,ajJ, lal = ao +...+an'
m6i cijp chi s6 i, a trang do 0


:::; n

x = (xo,...,xn), xa = x~o...x~n. Vdi

va lal = di, ta thift ifjp bifn

Ui,a tl1ang

ang. Khi do cha da thac P E Z[Ui,a]'dijt P(Fo,...,F;J ia gia tri co dl1(1cbdng cach
thay cac bifn Ui,a trang P biJi cac h~ stf Ci,atl1ang ang cua cac da thac Fa,..., F;!.
Nfu k ia trl1dng dong d{Ji stf thi tan t{Ji duy nhttt da thac Res E Z[Ui,a]thoa Dinh

iy 2.4 va dl1(1CgQi ia kit thac cua cac da thac Fo,...,F;t' ky hi~u Res(Fo,...,F;')'
Dinh Iy 2.4. Cha k ia trl1dngdong d{Jis6va Fa,...,F;t E k[xO,...,xn] ia cac da thac
thudn nhttt bgc do,"" dn tl1dng ang. Khi do tan t{Ji duy nh{{t da thac Res E Z[u"a]

co cac tfnh ch{lt sau:
(1)

V(Fo,...,F;J

+=

0 nfu va chi nfu Res(Fo,...,F;J = O.

(2) Res(x~, ..., x~n)= 1.
(3)

Res bttt khd quy tren Z[Uia].
,

Chung minh: xem [25, mve 78].

KSt thue du'Qe su dvng d€ khu cae biSn tITnhiSu phu'dng trinh. Vi v~y no


13

xua't hi~n cac diSm co sd thi h~ cac phuong trinh

Fo - xotgo =

... =

F;n - xmtgm

= 0 cling c6 nghi~m kh6ng tgm thuong Hi (0,,\,..., \n), khi d6 k€t thuc se tri~t
tieu ba't cha'p xo,...,x1ll'

2.1.2 Cd sd Grobner va bai tmin ifn boa
Chung ta b~t dgu voi mQt v~ti khai ni~m v€ da thuc nhi€u bi€n. MQt da
thuc f E k[xp..., xn] c6 thS duQc bi€u di6n bdi t6 hQp tuy€n Hnh hii'll h:~m
f
'
trong d 0 a

-

=

I:aaxa ,

aa E k

a

'7171

,

(a1,...,an ) E fL.J~ova

d'
x - (X1,...,Xn ). KhlOX
'

a

-

al

an

- Xl "'Xn

.

duQC gQl

la mQt don thuc trong k[xp""xn] va aaxa duQc gQi la mQt s6 h:~lllgcua da thuc

1. Khi lam vi~c voi vanh da thuc nhi€u bi€n, sv s~p x€p cua cac s6 h~ng d6ng

mQt vai tro quail trQng, do d6 chung ta cgn xac dinh mQt thu tv s~p x€p cac don
thuc cho cac don thuc tren k[X1,...,XJ
Djnb ngbia 2.6. M(jt tha tif dan thac tyen k[xp...,xn1La

m(jt quan h~ >--tyen Z~o

thoa cae tinh chitt sau:
(1)

>--La tha tif roan phan tyen Z~o'

(2)

Ntu

(3)

>--La tha tif t6t tYen Z~o'

a >--{3 va 'y E Z~o thi a + 'y >-(3+ 'y.

Nhu: v4y V(ji mtJi cijp dan thac xa va x!3 chi co m(jt tyong ba tyu:lJngh(fp sau xdy
ya La xa

>

x!3, xa = x!3 va xa

< x!3.

Djnb ngbia 2.7. Cho da thac f E k[xp-..,xrJ khac da thac khong va
tif dan thac.

>--

La m(jt tha


14

(1) Dan thac chil &;10cila f, ky hi~u LM(J), la dan thac XCVlcJn nha't rhea quan

he.

>-

saD cho acv

-:;C.

O.

(2) H~ s6 chil d(lo cila f, ky hi~u LCM(J), la h~ s6 aa cila dan thac chil d(lo.
Nlu LCM(J) = 1 thi f dl1(fc gQi la dan khai.
(3) 56 h(lng chil d(lo cila f, ky hi~u la LT(J), la sf;'h(lng ang wJi LM(J).
Dfnh nghia 2.8. M(jt tha tif dan thac tren k[xl,...,xl,xl+1,...,xn] dl1(fcgQi la m(jt
tha tif khil dfJ'i vdi cac biln

Xl'...,Xl nlu vdi mQi da thac 9 E k[Xl""'Xn]

va

LM(g) E k[Xl+W"'X,J thi 9 E k[Xl+1""'X,J.

Dfnh nghia 2.9. Cho ideal 1= (h""'fs) C k[xl""'x,J va 0::; l ::; n, ideal khil tha
l cila I la ideal II
II

C

k[xl+l,...,xn] dl1(fc xac dtnh bai

= In k[xl+W..,xn]'

Voi cac khai ni~m tren ve cac da thuc nhieu bie'n, chung ta se tlm hi~u cd
sd Grobner va cac ke't qua ve ly thuye't khii' bie'n.
Dfnh nghia 2.10. Cho trl1dcm(jt tha tif dan thac va ideal Ie k[Xl""'Xn], Tqp h(fp
cac da thac G = {gl,...,gJ C I dl1(fcgQi la ca sa Grobner cila I nlu
(LT(gl),...,LT(gs))

= (LT(I))

vcJiLT(I) la tqp h(fp cac s6 h(lng chil d(lo caa cac da thac thu(jc I.
Cd sd Grobner la mQt cong c\l tinh toan duQc ap d\lng d~ giai quye't kha
nhieu lOp bai toan nhu cac tinh toan trong ly thuye't ideal, tim nghi~m cua h~
phudng trinh da thuc, tinh toan cac syzygy, tim bi~u di~n fin cua d~ng tham so' da
thuc va hUll ty. Cac tinh cha't va thu~t toan xac dinh cd sd Grobner co th~ tham
khao chi tie't trong [9]. Ke't qua quail trQng sau day cua ly thuye't khii' cho phep

15

ta giai quySt bai taan fin hoa d<;lngtham s6 hUll ty.

Dfnh Iy 2.11. Cha sO'nguyen

a<1< n

va ideal 1 c k[xp...,xnJ. GQi G la cd sa

Grabner cua 1 ling vcJi mqt tha t1l khit dO'i vcJi cac bitn Xl',.., Xl' Khi d6 tfjp h(lp
cac da thac G[ = G n k[xl+1"",XnJdU:(lcgQi la cd sa Grabner cua id~al khit tha 1
cua 1,
Chung minh: xem [9, tr. 113],
Dfnh Iy 2.12. (Dfnh Iy dn hoa tham sf{ hoa hun ty) Cha V la da tqp c6 bi!u

diln tham sO'(1.1). GQi 1 la ideal dU:(lcxac dink bai 1 = (~Xl
l-yh)

C k[y,tp...,t""xp'",xnJ,

-

91,...,hnxn - 9n'

trang d6 h = ~h2...hn va 1"'+1= 1nk[xp"',x,,]

la ideal khil tha m + 1 cua 1. Khi d6 V(J",+J la da t(lP nhJ nhdt trang k" chaa

V.

Chung minh: xem [9, tr. 130].
Nhijn xet: Vi~c sa dvng biSn b6 sung y khi xay dl,1'ngideal cua d<;lngtham s6
dam baa rang trong tru'ong h<;lpdi€m cd sd xua't hi~n bai taan v~n co th€ giai
du'<;lc.

2.2 Syzygy va bai tmin dn hoa
Chung ta tlm hi€u m6i lien h~ giua cac syzygy va bi€u di~n tham s6 hUll
ty (1.1) thong qua cac khai ni~m sau.
Dfnh nghia 2.13. Cha cac da thac fi.,..., Is E k[xp..., xrJ, khi d6 mQt vectd da thac
(~,...,hJ E e[xp...,x,J

du(lc gQi la mQt syzygy cua fi.,...,1s ntu

~h +... + hsls = o.
Tfjp h(lp cac syzygy cua fi.,...,1s ky hi~u la Syz(fi.,...,Is).


16

M~nh d~ 2.14. Syz(A,...,fJ

fa mi}t k[xp""xn] -modun va dll(je gQi fa modun sy-

zygy eila A,...,!s.

Chung minh: xem [10, tr. 189].
Dfnh nghia 2.15. FJa tc}-pdi di}ng foC}-id va b4e bi}i (u1"",um)dll(je ~ae dtnh nhll
sau
Ul

Urn

~
. +4 ... tirn=
D xa ~D ... ~
D a?'
ZJ.".Zm'1
m
l"'l~d ~=o
im=O
'

trong

d 00:=

0'

(2.1)
'

fuJn '"
"'1
'"
,x
=X1 ...xnn.

'

/.

Kh 1 d OVO'lmOl

(O:l"",O:n ) El'1

;::.

" .,
b Qgzatn

.

(t1"'" tm )

ta xae dtnh dll(je mi}t biiu diln an eila mi}t da thue trong e.
Dfnh nghia 2.16. Xet da tc}-pV co biiu diln tham so' (1.1) trong trllong h(jp go :=

~ = ... = hn. Khi do ta noi da tc}-pdi di}ng (2.1) sinh ra V ntu vdi mQi bi} giG trt
(t1,...,tm) eila tham so'thz
+

F(g "1"'"

~
t m ) ..- D
l"'l~d
"'1

Ul

ga ~

D

~=0

Urn

... ~

D

'"

(2.2)

irn=0

gl(t1,...,tm)
gn(~,...,tm)
trong d a g =
...
[ go(t1"..,tm)]
( %(tp ...,tm)]
'

+~.. . t irn - 0
a a~."'i".'1
m ,
an

.
Ul

Til Dinh nghla 2.16 ta thty ding hQ cae da thuc { ~"'i~O

Urn

a:...imt;I...t;~.

}

la cae

syzygy cua cae da thuc biSu di~n d~ng tham s6 cua da t~p V.
Dfnh nghia 2.17. HQ cae da tc}-pdi di}ng ~, ...,F:. dll(je gQi fa di}e f(jp tuytn tinh
tren k[t1,...,tIn]ntu t6 h(jp tuytn tinh A~ +... + !sF: = 0 khi va chi khi A,...,Is E
k[t1,...,tm] fa ededa thTiekhong.

Phu'dng phap £n hoa sa d\mg cae da t~p di dQng, e\1thS la cae du'ong eong


17

va m~t di dQng, du'QCde xua"tvao 1995 bdi T. Sederberg va F. Chen [23]. Cho
hai du'ong thiing di dQng, khi ghi tri tham s6 t thay d6i thi cac giao diem cua
chung ve ra mQt du'ong congo Vdi ba phiing di dQng va khi gia tri tham s6 (8,t)
thay d6i thi giao cua chung ve ra mQt m~t.
Tuy nhien m6i lien ht%giua da t~p di dQng va syzygy l~i du'Qcnh~n d~ng
bdi D. Cox va du'Qc6ng tdnh bay mQt cach hoan chlnh va co ht%'th6ng b~ng

ng6n ngu cua d~i s6 giao hoan trong [7].
2.2.1 Syzygy va phu'dng trinh du'Xet bai toan in hoa du'ong cong tham s6 huu ty phiing (1.2). Khi do ta co
n = 2, m = 1 va cac da t~p di dQng (2.1) du'QcgQi la cac du'ong cong di dQng
b~c Ul'

Dfnh nghia 2.18. Nfu d

=1

thl dlldng Gong di dQng dll(/C gQi fa dlldng thang di

dQng b(jc Ul va co phllClng trlnh biiu diln

L(x, y, t) := A(t)x + B(t)y + C(t) = 0,
trong do A, B, C E k[t] fa cac da thac co b(jc tffi da fa Ul' f)lldng thang di dQng
dll(/C noi fa sinh dudng Gong tham sff (1.2) nfu

A(t)a(t) + B(t)b(t) + C(t)c(t) = 0
vdi mQi t.
Tu'dng tlj xet bai toan in hoa m~t tham s6 huu ty (1.3). Khi do ta co
n = 3, m = 2 va cac da t~p di dQng (2.1) du'QcgQi la cac m~t di dQng song b~c
(Ul,U2) ne'u m~t huu ty la tich tensor (m~t di dQng b~c U = Ul +U2 ne'u m~t huu

ty la phan tam giac). Cling nhu' d6i vdi tru'ong hQp du'ong cong, ta co cac dinh
nghla cho tru'ong hQp m~t.


18

Dfnh nghia 2.19. Ne'u d

=

1 cac mift di dqng du(lc gQi fa cac phdng di dQng song

bqc (Ul' U2) ne'u mift hClu ty fa tfch tensor (phdng di dQng bqc U =

Ul

+ U2 ne'u

mift

hCluty fa phan tam giac) va co phuang trinh bilu di8n
S(x,y,z,s,t):=

A(s,t)x + B(s,t)y + O(s,t)z +D(s,t)

= 0,

trong do A, B, 0, DE k[s,t] fa cac da thac song bqc (Ul,U2)ne'umijt hCluty fa tich
tensor (cac da thac bqc U =

Ul

+ U2 ne'u mift

hClu ty fa phan tam gia~). Phdng di

dQng du(lc noi fa sinh mift tham sri (1.3) ne'u
A(s, t)a(s, t) + B(s, t)b(s, t) + O(s, t)c(s, t) + D(s, t)d(s, t) = 0
wli mQi (s, i).
Tli Dinh nghla 2.18 ta tha'y Syz(a, b,c) la hQ cac duong th~ng di dQng sinh

du'ong cong tham s6 (1.2). Tu'dng tv voi Dinh nghla 2.19, Syz(a, b,c,d) cling la hQ
cac ph~ng di dQng sinh m~t tham s6 (1.3). Hdn mla Syz(a, b,c) va Syz(a, b,c,d) la
cac modun syzygy co tinh cha't d~c bi~t sau.
Mc$nh d~ 2.20.
(1) Ne'u a, b, c fa cac da thac trong biiu di8n d(;mg tham so' cila du?Jng Gong thi
Syz( a, b, c) fa modun t1/ do.

(2) Ne'u a, b, c, d fa cac da thac trong biiu di8n dr;mg tham so' cila mijt thi
Syz( a, b,c, d) fa modun t1/ do hqng 3.

Chung minh: xem [11, tr. 812] va [1, tr. 704].
Cac ke't qua chinh trong [7, 8, 11, 23] kh~ng dinh trong truong hqp duong
cong va m~t tham s6 thong co diem cd sa, ta co the tlm duqc cac du'ong th~ng di
dQng va cac ph~ng di dQng (ho~c m~t b~c hai di dQng) dQc l~p tuye'n tinh sinh
du'ong cong va m~t tham s6 hii'llty.


19

Phudng trlnh in cua duong cong tham sO'hUllty (1.2) co the duQCxac dinh
bdi u duong th&ng di dQng b~c u -1 dQc l~p tuytn Hnh co d~ng
u-I

A(t)x + Bi(t)y + Ci(t) = L: Lij(X,y)tj,
j=O

trong do Lij(X,y) la cac h~ sO'tuytn tinh theo x, y. Dinh thlic cua cae h~ so nay
ILij(X,y)1= 0 Ia phudngtrlnhin cua duongcong thamsO'[7,23].

Tudng tv truong hQp duong cong, phudng trlnh in cua m~t tham sO'hUll ty
tich tensor (1.3) co the duQCxac bdi 2uv ph&ng di dQng song b~c (2u -1, v-I)
dQc l~p tuytn tinh co d~ng
2u-I v-I

A(s, t)x + Bi(s, t)y + Ci(s,t)z + Di(s,t) = L: L: ~jk(X, y, z)sjtk,
j=O k=O

trong do ~jk(X,y, z) Ia cac h~ s6tuytn tinh theo x, y, z. Dinh thlic eua cac h~ sO'
nay l~jk(X, y, z)1 = 0 la phudng trlnh in eua m~ttham sO'[7, 8].
Nhu v~y de ap dl;1ngphudng phap cac duong th&ng va cac ph&ng di dQng,

chung ta cgn phai giai h~ 2u phudng trlnh tuytn tinh voi 3u ttn de tlm cae h~ so
cho cae da thlic Ai(t), Bi(t), Ci(t) va h~ 6uv phudng trlnh tuytn tinh voi 8uv ttn de
tlm cac h~ sO'cho cac da thlic A(s, i), Bi(s, i), Ci(s, i), Di(s,t) tudng ling.
2.2.2 p,-cd sd cua du'(jng cong va mi)t tham s6 hUll ty
Voi nh~n xet t~p hQp cac duong th&ng di dQng sinh duong cong tham sO'
hUll ty va t~p hQp cae ph&ng di dQng sinh m~t tham sO'hUll ty la cac modun
syzygy

tv do, chung ta co the giiU quytt bai toan in hoa thong qua mQt cd sd

d~c bi~t, duQc gQi la j.L-cdsd cua duong eong ph&ng va m~t tham sO'hUll ty, cua


20

cac m6dun nay.
Dinb ngbia 2.21. Cho mQt vectClda thac p E 1R[t]k dl1(/c biiu diln nhl1 sau
m

P = (Pl(t),P2(t)"",Pk(t)) = 'L)Pil,Pi2"",Pik)e,
i=O
vdi vectCl h~ so' chu d(LO (Pml,Pm2,...,Pmk) +=O. Ta ky hi~u:

(1)

VectClchu df,lo cua vectClda thac p ta LV(p) = (Pml'Pm2,...,Pmk)'

(2) B{jc cua vectClda thac p ta deg(p)

= m.

Dinb ngbia 2.22. Hai dl1iJng thdng di dQng p

= PI (t)x + P2(t)y+P3(t) va q = qI(

t)x + q2(t)y + q3(t) dl1(/CgQi ta jL-CClsa cua dl1iJngGong tham so' hilu ty phdng
(1.2) ntu:
(1) P va q ta cClsa cua Syz(a,b,c), nghia ta vdi mQt dl1iJngthdng di dQng bat ky
L E Syz(a, b,c) dl1(/c biiu diln duy nhat nhl1 sau
L

voi ~,~
(2)

= ~P + ~q,

(2.3)

E 1R[t].

P va q co b{jcnho nhat so vdi tat cd cac cClsa cua Syz(a,b,c).
Sv t6n t~i cua jL-cd so cua duong cong thalli sa huu ty ph~ng (1.2) va cac

tinh cha't cua jL-cd so duQc trinh bay trong [9]. Tinh chit quail trQng san day ap
dl;lngde' giai quye't bai toan in hoa duong cong thalli sa huu ty ph~ng.
Dinb ly 2.23. Cho P = PI(t)x + P2(t)y+ P3(t) va q = qI(t)X+ Q2(t)y+ q3(t) ta mQt
jL-CClsa cua dl1iJng Gong tham so' hilu ty phdng (1.2), vdi deg(p)

< deg(q). Khi do

phl1C1ngtrlnh an cua dl1iJng Gong tham sf;' (1.2) dl1(/c xac dinh bai Res(p, q,t) ta


21

kit thac cila p va q theo t.
Chung minh: xem [2, tr. 372].
Binh nghia 2.24. Ba phdng di dQng p

= Pl(S, t)x

ql (s, t)x + q2(S,t)y + q3(s, t)z + q4(s, f), r

=

+ P2(S,t)y + P3(S,t)z + P4(S,f), q =

7i(s, t)x + r2(s,t)y + r3(s,t)z + r4(s, t) sinh
~

m(it tham s5 hflu tY (1.3) saG cho
(2.4)

[p,q,r] = K,(a,b,c,d)

trong d6 K, la hling s5 khdc khong. Khi d6 p, q, r duC/cgqi la It-c(J siJ cila m(it tham
s5 (1.3). [p, q,r] duC/cgqi la tich ngoqi cila P, q, r va duC/cxdc dinh biJi
P2

P3

P4

[p,q,r] = IIq2

q3

q4'-

r2

r3

r4

1

I

Pl

P3

P4

Pl

P2

P41

ql

q3

q4'11 ql

q2

q41,-lql

7i

r3

r4

r2

r4

7i

IPl

7i

P2

P3

q2

q3'"

r2

r3

Binh nghia 2.25. p, q, r duC/cgqi la It-c(J siJ to'i tilu cila m(it tham so' (1.3) tich

tensor (phan tam gidc) niu p, q,r thoa (2.4), degt(p) + degt(q) + degt(r) la nho
nhdt va degs(p) + degs(q) + degs(r) la nho nhdt (niu P,q,r thoa (2.4), deg(p) +
deg(q) + deg(r) la nho nhdt).

Sv t6n t~i cua It-co sa cua m~t thalli s6 huu ty (1.3) va cac tinh cha't cua
no duQc trlnh bay trong [1]. K€t qua sau cho chung ta tha'y co the ung d\lllg It-co
sa de xac dinh phuong trlnh £n cua m~t.
Binh ly 2.26. Cho If = (dx - a,dy - b,dz - c,dw -1)

n JR[x,y,z,s, t] la mQtidealva

g(s) E JR[s]la da thac xdc dinh biJi (a,b,c,d) n JR[s] = (g). Khi d6 If la ideal nguyen
t5 va g(s) tf-If. HcJnnfla, f E If niu va chI ne'u f = 0 la mQtphdng di dQng sinh


22

m(it tham sff hilu ty (1.3). Cl;lthi, ntu f(x, y, z) = 0 la phu:ang trtnh an cila m(it (1.3)

tht f(x, y, z) E II.
Chung minh: xem [1, tr. 696].
Dfnh Iy 2.27. Cho ]'va g(s) nhu: trong Dinh If 2.26 va p, q, r la J.1,-Cd
~a caa m(it
tham so' hilu ty (1.3). Khi do
00

]' = (p,q,r): gOO= U (p,q,r): gN =
N=O

{f gNf E (p,q,r),N > a}.
I

Ntu t6t cd cac ddm cd sa hilu hc;mcaa m(it tham sff hilu tY (1.3) lil giao day dil dia

phu:ang tht
]'

= (p,q,r).

Chung minh: xem [1, tr. 696].