第一届全国计算机数学学术会议报告摘要

z:(1)(2)(3)zAdeterminedalgorithmforirreducibledecompositionofalgebraicvarietiesandprimarydecompositionofzero-dimensionalideal.ThisisajointworkwithDingkangWangandFushengLeng.LetFbeapolynomialsetin,wecandecomposeFintoaseriesofascendingchain{}givenbypolynomialsirreducibleoverQfirst.Thentogetirreducibledecomposition,wemustdecomposeeveryintoirreducibleascendingchains.Letbeanascendingchainwithbetheleadingvariableof,theothervariableswillbedenotedby1[,,]mQxxKiAiA1{,,}nAPP=KiyiP1,,suKu,sos+n=m.Let1(,,)sKQuu=K,thenAiszero-dimensionalascendingchainoverK.LetdbethenumberofsolutionsofA.Substitute122nnyyayay=++KintoA.Weprovethatamong21()ndC−integralvectorsin(n-1)-cubewithinteger,theremustbe2(,,)naaK221122[,1][,1][,1]ddnnbbCbbCbbC+−×+−××+−K2dib1onesuchthatallascendingchainswithpolynomialsirreducibleoverQfortheorderareeitherwithlengthnorintheforms2(,,)naaK2nyyyK1221(,,,)(,)()nnQyyyQyyQyKK222222(,)()(())nQyyayygy=+K2(,,,)()(())iniiiiiQyyyayygy=+KThesenewascendingchainsgiveirreducibledecompositionofA.Besides,wecanuseonlyalsosomespecial’stoarriveatthesameaim.Ouralgorithmisusedalsotogiveprimarydecompositionofzero-dimensionalideal.2(1)dnC−+12(,,)naaKzComputationofMeshSurfaceswithPlanarFacesIshalldiscusstheproblemofrepresentingafree-formshapebyameshsurfacewithplanarquadrilateralorhexagonalfaces.Thisproblemismotivatedbytheneedinarchitecturefortilingfree-formbuildingsurfaceswithplanarglasspanels.Severaleffectivemodelingmethodswillbepresentedbasedonsomenovelconceptsfromdiscretedifferentialgeometry,includingconicalmeshesandDupinduality.Ishallalsodiscussthecomputationofoffsetandcurvatureofthesediscretesurfaces,andtheirconnectionstoshapemodelingofdiscreteconstantmeancurvaturesurfaces.JointWorkwithYangLiu,HelmutPottmann,JohannesWallnerandAlexanderBobenkozLeibnizDescartes“”“”Grassmann-CayleyGrassmann-CayleyClifford“”2CGAGrassmann-CayleyCliffordGrassmann-CayleyspinLiespin1970RotaNBAGrassmann-CayleyNGCNGAGrassmann-CayleyzAES128192256AES-128AES-192AES-256AESAESAES-128H.GilbertM.Minier7,2128.N.FergusonB.Schneier7Square,2128-2119..AES-128,2115211912HASH3AESARIA2004KSX1213ARIA128128\192\256,3121416.ARIASP,P,PARIA,ARIAARIA4ARIA46ARIAP3232Camellia8CamelliaCamellia192/25612Camellia8Camellia41NewtonNewton,lowerLagrange,Hermite,Newtonlower.,tower,LagrangeNewton.Birkhoff.Hermite,BirkhoffNewton.BoYuAndBoDongDepartmentofAppliedMathematics,DalianUniversityofTechnology,Dalian,Liaoning116024,China.([e-mail]).DepartmentofAppliedMathematics,DalianUniversityofTechnology,Dalian,Liaoning116024,China.([e-mail]).ASymmetricHomotopyAndHybridMethodForSolvingMixedTrigonometricPolynomialSystemsPolynomialsystemscomingfrommixedtrigonometricpolynomialsystemshaveaspecialstructure:thelastmequationsarex2n+i+x2n+m+i¡1=0;i=1;:::;m.Andthemadditionalquadraticequationshaveaninherentsymmetry.Inthispaper,exploitingthespecialstructureandthesymmetry,asymmetrichomotopyisconstructedand,combininghomotopymethod,decompositionandeliminationtechniques,ane±cienthybridmethodforsolvingthisclassofpolynomialsystemsispresented.Usingthenewhybridmethod,someproblemsfromtheliteratureandachallengingpracticalproblemaresolved.Numericalresultsshowthatourmethodismuche±cient.Keywords.polynomialsystem,mixedtrigonometricpolynomialsystem,homotopymethod,hybridalgorithm.LiuJinwang,lidongmei,Fuxiaoling(CollegeofMathematicsandComputation,HunanScienceandTechnologyUniversity,Xiangtan,Hunan,411201,Chinae-mail:[e-mail])ThetermorderingswhichareHomogeneoulyCompatiblewithCompositionLetK[x1,…,xn]bethepolynomialringoverafieldKinvariablesx1,…,xn.Let),,(1nθθL=ΘbealistofnhomogeneouspolynomialsinK[x1,…,xn].PolynomialcompositionbyistheoperationofreplacingxiofapolynomialbyΘiθ.Wesaythat5compositionbyishomogeneouslycompatiblewiththetermorderingifforalltermspandq,pqdegp=degqimpliesthat)()(ΘΘΘltqltpoo.Howtotestitisverydifficult,inthispaper,weshallobtainadecisionprocedurefortestingit;andobtainsomeimportantproperties:Proposition1eequivalentdegFollowingsar(i)p=degq;,,qp∀∀)()(ΘΘ⇒qplpqlppoo;(ii)yZynrr,∈∀ishomogeneous,0011⋅⋅⇒⋅ATyAyrr.Proposition2Folllentishomogeneous,owingsareequivayZynrr,∈∀0011⋅(i)⋅⇒⋅AyAyTrr;11⋅AT(ii)0,⋅⋅⇒⋅⋅∈∀MyAMyZynrrr0.Proposition3Followingsareequivalent;11⋅⋅⋅ATMB00,11⋅⋅⋅⇒⋅⋅∈∀ATMyAMyZynrrr(i)(ii)0,⋅⇒⋅⋅⋅∈∀yAMByZynrrr0.Proposition4Followingareequivalent11⋅⋅⋅⋅⋅ATMDAM(1)0,⇒⋅⋅∈∀yDyZyn0rrr;(2)thestandardform()QPSM=ofM)(ATMA⋅⋅⋅Misarystepmatrix.Propositlowingareeqent(1)degp=degq;abinion5Thefoluival,,qp∀∀)()(ΘΘ⇒lpqlppqpoo;of)(QPSM=)(ATMAM⋅⋅⋅M(2)thestandardformisabinarystepmatrix.cangivemoreationonpolynomialideal,suchas,dimension,maximumindependentset,Hilbertnomial,etc.Butinthistalkwepresentonlyaneigenvaluemethodtofindsomecomponentsstems,whichisbasedontheborderbases.ChenYufuGraduateUniversityofChineseAcademyofSciencesBorderBasesforPositiveDimensionalPolynomialSystemsFortheresolutionsofzerodimensionalpolynomialsystems,eigenvalueandeigenvectormethodsareeffective.Thereborderbasesareneededandanefficiencyalgorithmtocomputeaborderbasisisimportant.Inthistalkwediscusshowtogeneral