无限维空间中的实零点定理

1、39319965ACTAMATHEMATICASINICAVol.39,No.3May,1996(510400)(330047):a,aRR,A;KXA,(K,P)XAARA=()|AKn|A|AA:=An,AnnA|A|=|A|.nKXJJKXfiKX,piP.fiJ;JnfiKX,piP,pifi2i=1m 2 JS=,S:=1+pifi mi=1IKXURAVR(I)=()RA|JR(U)=fKX| R=fKX|AfI,f()=0;()U,f()=0;m,fiKX,piPf2m+pifi2I.JR(VR(I)=R,D.W.Dubois219691970AS.Lang4Hilbert199

2、452433371A1()(S1)(S3)IKXI(S2)IJR(VR(I)=KXRVR(I)=.()RAA.MMM=JR().(S4)AMKDD/MLD=K(K,P)K(S5)(S1)(S2):JR(VR(I)RD(S4),KDL“”.fJR(VR(I),t,KXtVR(I )=.S1pjPpjgj.t=1/f,fih i+(1tf)h=1+RfI =(I,1tf),fiI,h i,h,gjKXtff2n(S2)(S3):1B/CIBIBICCR/KARR/KRRKKKrR(S2)(S3)()RAM=J(),MKXXARX(X)KX=RX/(X)(X)RX1JR().=R2JR()KXKXM

3、KX(1)=M=R=JR(VR(M),VR(M)=.(X)()VR(M),JR()M.JR()MM=JR(),(S3)(S4):()RA.IK-KXKKMIKXMM /I=MKXKX(S3)KXKD/MR,()RA,M =JR().(S2)(S3)D/M=K=KX/M R/KD/M(K,P)M(S4)(S5):KX/IK.IK-IKXKK-:KXZornKXJI.33839(S4),E=KX/J(K,P)PP(E,P )(K,P),E(K)L(K),(P)5(3.9)ZornDKX/IKX/J=EL,1Ef+If+J.I=|E··1,IS=,DL|K=,(S5)(S1):V

4、R(I)=.IKXJS=,JIIS=.QZornJ|JKXQQnpiP,fiKXQ,f=n i=1pifi2Q.(1)m(1)(Q+f1·KX)S=.mgQ,h,hiKX,qiP,g+hp1f1=1+qih2i.i=1n 2g(g+2hp1f1)=1+rjgj+h2p1f+pifi2,rjP,gjKX.Q+f1·KX Q.Qg(g+2hp1f1)+hp1f=1+22rjgj+ni=2i=2(p1pi)(hfi)2QS,:=X+Q.QQKX/QK(S5)K(K,P)RK-.()RAQIVR(I)=.(K,P)|A|1)(K,P)|A|32D,DD /DDD(K,P)|A|(i

5、)A(ii)Atr.dK=b,b>a,AaAtr.dKKQ(i)(ii)ba.A(K,P)RCca(RRb).CcR=|C.YR(Y)22 =Y1/(Y)|C1(Y)+,C.3339c,RYDR(Y).,fDC=|C.D=R,RRY(Y)(2(Y)2+)f1D,D=R(Y)2DK3KR(K,P)KR(Y)RR(Y)(K,P)KKR(Y),K(K,P)(0)YK(=K|C)KK/(0)=K(S4)(K,P)2(ii)4K=R=R(QQ(T),A=N(),tr.dR>|N|.tr.dR|N|,RRT,|T|N|,TQ(T)RX1,···,Xj,·

6、··I,2(X1j)Xj1,j=2,3,···.I1+M j=2m i=1fi2I,fiRX1,···,Xj,···,M,1+fi,gjRX1,···,XM(m i=1fi2=gj2gj(X1j)Xj1,).(2)j=2,3,···,M.(2)Xj=j(j=1,2,···),1=M+1,j=m 1+fi2(1,···,M)=0,i=1IVR(I)=.(a1,·

7、;··,aj,···)RI0,j=2,3,···,(a1j)a2j=1>0,(a1j)a2j1=a1j>0,(S1)a1>j,j=2,3,···.fKX,ZariskiR1|A|()RA|f()=0;VR(f)=()RA f()=0.(K,P)DR(f)=DR(f)DR(g)=DR(fg),RADR(f)|fKXZariskiRAVR(f),fKX,(K,P)|A|A|3(K,P)RAZariskiF|RAAF=.RAF=VR(f),f|fKX,.JKXVR(J)=V

8、R(f)=.1(S1)JKXm,npiP,hi,gjKX34039fj1+mi=1pih2i=n j=1fjgj.nn(3)R(3)=VR(f1,···,fn)=VR(fj)=Fj,j=1j=1RA1(S1)IKXVR(I)=.RAVR(f)|fI,fIVR(f)=VR(I)=.RAf1,···,fsI=VR(fi)=VR(f1,···,fs).i=1s(4)I MfiKX1,···,XM,i=1,···,s.(4)f1,··

9、·,fsKX1,···,XMMRMf1,···,fsI IgiKX1,···,XM,piPm,KX1,···,XMm 21+pigiI I.IKXi=1(S1)2RAZariski1Ru,v,u<v,u,v,vu,+xR|fi(x)>0,i=1,···,m;g(x)=0,()fi,gKX,i=1,···,m.u,+R(X)uK 22···(Xvt)2+wt,(X)=

10、(Xu1)···(Xus)(Xv1)2+w1,vui,vj,wjR,wj=0,i=1,···,s;j=1,···,t.u1<u2<···<us,u=ur,1rs.(Y,X):=Y2(s+2t)·(X1/Y2)=si=1t 224(Xui)Y1·XY2vjY21+wjYKY,X.2j=13341xu,+,(Y,x)2rxKY1,···,Y2r(Y1,x)=0···(Y2r,x)=0

11、Y1<···<Y2r).(55.1),()(Y,x)=0RR2rx2u,v,u=,v=+,m,u,vR1+mK-R1,u,v()1>0,m.()R1+mRK-R1+mV:fi(X)Yi21=0,i=1,···,m;g(X)=0.R(x,y1,···,ym)xV().3A),A,S=S,k,nN,k=1nSA(A,r,1n,S,rA(A ,r)A ArA NA ,1rn,S,rA ,SAAA.k=1,···,nS,kS=()Sb,1bn,S,bS=A0=,r=

12、b.(A0,r),“ ”(A1,r1),(A2,r2),(A1,r1) (A2,r2)Zorn(A,r).A1A2,r2A1r2|A1=r1.A=A.AA,S,rASAAbN,1bn,A =A.r A N(A ,r ),(A,r)S,rAS,bSAA r=b,r=r,A.A=A,41,(i)(K,P)|A|(ii)R(iii)RAAAA(i)(ii).TARA2A,VmN,TR1+mK-VRKX,X,1,···,X,mI34239X,k,=X,k=X,k,k=1,···,m;k=1,···,m.XX,k(

13、A,k=1,···,m)I(A)|A|,XA.IIKX,X,k|A,k=1,···,mn 1+pifi2I,piP,i=1fiKX,X,k|A,k=1,···,m,stjj=1rj=11,···,sA,1+n i=1pifi2=gj,rjhj,rj.hj,rjIj,gj,rjKX,X,k|A,k=1,···,m,j=1,···,s;xTj.j=1srj=1,···,tj.TATjVjRV

14、j(x,xj,1,···,xj,mj),j=1,···,s.i=11+InX=x,Xj,kj=xj,kj,j=1,···,spifi2(·)=0.kj=1,···,mj,(i)(,k|A,k=1,···,m).A,(,1,···,m)V,(ii)(iii).RT.T.(ii)ASAR12.14SRTRTA3SA,TT,AT=T;1(ii)TASA(iii)(i).4(S1)IIKX|A408fKX|A,A.

15、()A()A ,(,A )X=(A )ARf(X=|A ),ARX|AA =.(,A )A.I=(1,···,s)A,X1,···,Xm,X,fkKX1,···,Xm,k=1,···,s.KX1,···,XmX1=X.f1,···,fs1KX1,···,Xmf1,···,fsVRm.SV(x1,···,xm)x1(iii)2.2

16、1SRS1S2S12,1,2A.SA|A|=|A|,AS=.bAS,A0=,=b.(,A0).R2f(X=)ARX|=R=(1,···,s)A,1+n i=12gi=s j=1hjfj(X=),gi,hjRX|=.3343S(a1,···,am)V,k=2,···,m,a1=b=.n 21+gi(·)=0.Xk=ak,i=1“ ”(1,A1),(2,A2),(1,A1) (2,A2)A1A2,2A12|A1=1.A=A.AA,RX /A(,A).fX= AARX /AfX=,A;X=bAbR,Z

17、ornA =A,AR=b,=,A.( ,A ),(,A)f()AR=RX|/ARf()=0,A,()AI4a,aAaµ,µµ|µ|aµ1,aµA,Aµ,=µ.A,u<µ. µ |a=a·a=a.µµ=µ.|µ|=|+1a,Fµ:FAAµ,µ,F1Qµ,<µ,(i)µFµ=F.(ii)µ<µµ=+1,.FAT:=tFFF(T).=u

18、,vA,F(T)tu,u=,u=1;v=vt,u,v,A,v=+,v=1.u=C() 2 ciwiciC,wiF(T).F(T)Sµ:=1/Sµ.F(T)SµPµ.Fµ(F(T),Pµ)µ,RFµ.RR:=Fµ,µRa=u,vARA,µ,u,vFµ,µ<,A.Fµ:=+1.Fµt,R34412345RealNullstellensatzinInnite-DimensionalSpacesZengGuangxing(DepartmentofMathematics&SystemScience,NanchangUniversity,Nanchang330047,China)LiangSongxin(DepartmentofMathematics,GuangzhouTeachersCollege,Guangzhou510400,China)Abstract:Inthispaper,weestalbishtherealNullstellensatzininnite-dimensionalspaces,andcharacterizethoseorderedeldsforwhichtheinnite-dimensionalre