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ID:34866659
大小:3.29 MB
页数:478页
时间:2019-03-12
《Springer.Galois.Theory.of.Linear.Differential.Equations.van.der.Put.M.Singer.M.[Springer.2003](478p)》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
1、GaloisTheoryofLinearDifferentialEquationsMariusvanderPutDepartmentofMathematicsUniversityofGroningenP.O.Box8009700AVGroningenTheNetherlandsMichaelF.SingerDepartmentofMathematicsNorthCarolinaStateUniversityBox8205Raleigh,N.C.27695-8205USAJuly2002iiPrefaceThisbookis
2、anintroductiontothealgebraic,algorithmicandanalyticaspectsoftheGaloistheoryofhomogeneouslineardifferentialequations.AlthoughtheGaloistheoryhasitsoriginsinthe19thCenturyandwasputonafirmfootingbyKolchininthemiddleofthe20thCentury,ithasexperiencedaburstofactivityinthe
3、last30years.Inthisbookwepresentmanyoftherecentresultsandnewapproachestothisclassicalfield.Wehaveattemptedtomakethissubjectaccessibletoanyonewithabackgroundinalgebraandanalysisatthelevelofafirstyeargraduatestudent.Ourhopeisthatthisbookwillprepareandenticethereaderto
4、delvefurther.Inthisprefacewewilldescribethecontentsofthisbook.Variousresearchersareresponsiblefortheresultsdescribedhere.Wewillnotattempttogiveproperattributionsherebutreferthereadertoeachoftheindividualchaptersforappropriatebibliographicreferences.TheGaloistheor
5、yoflineardifferentialequations(whichweshallrefertosimplyasdifferentialGaloistheory)istheanalogueforlineardifferentialequationsoftheclassicalGaloistheoryforpolynomialequations.Thenaturalanalogueofafieldinourcontextisthenotionofadifferentialfield.Thisisafieldktogetherwith
6、aderivation∂:k→k,thatis,anadditivemapthatsatisfies∂(ab)=∂(a)b+a∂(b)foralla,b∈k(wewillusuallydenote∂afora∈kasa).ExceptforChapter13,alldifferentialfieldswillbeofcharacteristiczero.Alineardifferentialequationisanequationoftheform∂Y=AYwhereAisann×nmatrixwithentriesinkal
7、thoughsometimesweshallalsoconsiderscalarlineardifferentialequationsL(y)=∂ny+a∂n−1y+···+ay=0(theseobjectsn−10areingeneralequivalent,asweshowinChapter2).Onehasthenotionofa“splittingfield”,thePicard-Vessiotextension,whichcontains“all”solutionsofL(y)=0andinthiscasehast
8、headditionalstructureofbeingadifferentialfield.ThedifferentialGaloisgroupisthegroupoffieldautomorphismsofthePicard-Vessiotfieldfixingthebasefieldandcommutingwiththede
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