[线性微分方程的伽罗瓦理论].van.der.Put.M.,.Singer.M.Galois.Theory.of.Linear.Differential.Equations.[Sprin.pdf

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1、GaloisTheoryofLinearDifferentialEquationsMariusvanderPutDepartmentofMathematicsUniversityofGroningenP.O.Box8009700AVGroningenTheNetherlandsMichaelF.SingerDepartmentofMathematicsNorthCarolinaStateUniversityBox8205Raleigh,N.C.27695-8205USAJuly2002iiPrefaceThisbookisanintroduction

2、tothealgebraic,algorithmicandanalyticaspectsoftheGaloistheoryofhomogeneouslineardifferentialequations.AlthoughtheGaloistheoryhasitsoriginsinthe19thCenturyandwasputonafirmfootingbyKolchininthemiddleofthe20thCentury,ithasexperiencedaburstofactivityinthelast30years.Inthisbookwepres

3、entmanyoftherecentresultsandnewapproachestothisclassicalfield.Wehaveattemptedtomakethissubjectaccessibletoanyonewithabackgroundinalgebraandanalysisatthelevelofafirstyeargraduatestudent.Ourhopeisthatthisbookwillprepareandenticethereadertodelvefurther.Inthisprefacewewilldescribeth

4、econtentsofthisbook.Variousresearchersareresponsiblefortheresultsdescribedhere.Wewillnotattempttogiveproperattributionsherebutreferthereadertoeachoftheindividualchaptersforappropriatebibliographicreferences.TheGaloistheoryoflineardifferentialequations(whichweshallrefertosimplya

5、sdifferentialGaloistheory)istheanalogueforlineardifferentialequationsoftheclassicalGaloistheoryforpolynomialequations.Thenaturalanalogueofafieldinourcontextisthenotionofadifferentialfield.Thisisafieldktogetherwithaderivation∂:k→k,thatis,anadditivemapthatsatisfies∂(ab)=∂(a)b+a∂(b)fora

6、lla,b∈k(wewillusuallydenote∂afora∈kasa
).ExceptforChapter13,alldifferentialfieldswillbeofcharacteristiczero.Alineardifferentialequationisanequationoftheform∂Y=AYwhereAisann×nmatrixwithentriesinkalthoughsometimesweshallalsoconsiderscalarlineardifferentialequationsL(y)=∂ny+a∂n−1y+··

7、·+ay=0(theseobjectsn−10areingeneralequivalent,asweshowinChapter2).Onehasthenotionofa“splittingfield”,thePicard-Vessiotextension,whichcontains“all”solutionsofL(y)=0andinthiscasehastheadditionalstructureofbeingadifferentialfield.ThedifferentialGaloisgroupisthegroupoffieldautomorphism

8、softhePicard-Vessiotfieldfixingthebasefieldandcommutingwiththede