Springer.van.der.Put.M.Singer.M.Galois.Theory.of.Linear.Differential.Equations.[Springer.2003](478p)

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1、GaloisTheoryofLinearDifferentialEquationsMariusvanderPutDepartmentofMathematicsUniversityofGroningenP.O.Box8009700AVGroningenTheNetherlandsMichaelF.SingerDepartmentofMathematicsNorthCarolinaStateUniversityBox8205Raleigh,N.C.27695-8205USAJuly2002iiPrefaceThisbookisanintroductiontothealgebra

2、ic,algorithmicandanalyticaspectsoftheGaloistheoryofhomogeneouslineardifferentialequations.AlthoughtheGaloistheoryhasitsoriginsinthe19thCenturyandwasputonafirmfootingbyKolchininthemiddleofthe20thCentury,ithasexperiencedaburstofactivityinthelast30years.Inthisbookwepresentmanyoftherecentresult

3、sandnewapproachestothisclassicalfield.Wehaveattemptedtomakethissubjectaccessibletoanyonewithabackgroundinalgebraandanalysisatthelevelofafirstyeargraduatestudent.Ourhopeisthatthisbookwillprepareandenticethereadertodelvefurther.Inthisprefacewewilldescribethecontentsofthisbook.Variousresearche

4、rsareresponsiblefortheresultsdescribedhere.Wewillnotattempttogiveproperattributionsherebutreferthereadertoeachoftheindividualchaptersforappropriatebibliographicreferences.TheGaloistheoryoflineardifferentialequations(whichweshallrefertosimplyasdifferentialGaloistheory)istheanalogueforlineard

5、ifferentialequationsoftheclassicalGaloistheoryforpolynomialequations.Thenaturalanalogueofafieldinourcontextisthenotionofadifferentialfield.Thisisafieldktogetherwithaderivation∂:k→k,thatis,anadditivemapthatsatisfies∂(ab)=∂(a)b+a∂(b)foralla,b∈k(wewillusuallydenote∂afora∈kasa
).ExceptforChapter13,

6、alldifferentialfieldswillbeofcharacteristiczero.Alineardifferentialequationisanequationoftheform∂Y=AYwhereAisann×nmatrixwithentriesinkalthoughsometimesweshallalsoconsiderscalarlineardifferentialequationsL(y)=∂ny+a∂n−1y+···+ay=0(theseobjectsn−10areingeneralequivalent,asweshowinChapter2).Onehas

7、thenotionofa“splittingfield”,thePicard-Vessiotextension,whichcontains“all”solutionsofL(y)=0andinthiscasehastheadditionalstructureofbeingadifferentialfield.ThedifferentialGaloisgroupisthegroupoffieldautomorphismsofthePicard-Vessiotfieldfixingthebasefieldandcommutingwiththede