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1、ONH.WEYLANDH.MINKOWSKIPOLYNOMIALSVictorKatsnelson∗06February2006ErasmDarwin,thenephewofthegreatscien-tistCharlesDarwin,believedthatsometimesoneshouldperformthemostunusualexper-iments.Theyusuallyyieldnoresultsbutwhentheydo....Soonceheplayedtrumpedinfrontoftulipsforth
2、ewholeday.Theex-perimentyieldednoresults.Weintroducecertainpolynomials,so-calledH.WeylandH.Minkowskipoly-nomials,whichhaveageometricorigin.Thelocationofrootsofthesepoly-nomialsisstudied.1.H.WEYLANDH.MINKOWSKIPOLYNOMIALS.LetMbeasmoothmanifoldofdimensionn:arXiv:math/0
3、702139v2[math.CV]14Feb2007dimM=n,whichisembeddedinjectivelyintotheEuclideanspaceofahigherdimension,sayn+p,p>0.WeidentifyMwiththeimageofthisembedding,soweconsiderMasasubsetofRn+p:n+pM⊂R.∗Departmentofmathematics,theWeizmannInstitute,Rehovot,76100,Israel.victor.katsnel
4、son@weizmann.ac.il,victorkatsnelson@gmail.com1Forx∈M,letNxbethenormalsubspacetoMatthepointx.NxisanaffinesubspaceoftheambientspaceRn+p,dimNx=p.Fort>0,letDx(t)={y∈Nx:dist(y,x)≤t},(1.1)wheredist(y,x)istheEuclideandistancebetweenxandy.IfthemanifoldMiscompact,andt>0issmall
5、enough,thenDx1(t)∩Dx2(t)=∅forx1∈M,x2∈M,x16=x2.(1.2)DEFINITION1.1:ThesetRn+pdef[TM(t)=Dx(t)(1.3)x∈MissaidtobethetubeneighborhoodofthemanifoldM,orthetubearoundM.Thenumbertissaidtobetheradiusofthistube.IsitclearthatformanifoldsMwithoutboundary,TRn+p(t)={x∈Rn+p:dist(x,M
6、)≤t},(1.4)Mwheredist(x,M)istheEuclideandistancefromxtoM.Thus,formanifoldswithoutboundary,theequality(1.4)couldalsobetakenasadefinitionofRn+pthetubeTM(t).However,formanifoldsMwithboundarythesetsTM(t)definedby(1.3)and(1.4)donotcoincide.Inthis,moregeneral,casethetubearou
7、ndMshouldbedefinedby(1.3),butnotby(1.4).HermannWeyl,[Wey1],obtainedthefollowingresult,whichisthestartingpointofourwork:THEOREM[H.Weyl].LetMbeasmoothcompactmanifold,withoutbound-aryorwithboundary,ofthedimensionn:dimM=n,whichisembeddedintheEuclideanspaceRn+p,p≥1.I.Ift>
8、0issmallenough1,thanthe(n+p)-dimensionalvolumeVoln+pRn+pofthetubeTM(t)aroundM,consideredasafunctionoftheradiust1Ifthecondition(1.2)issatis
