galois theory of linear differential equations - m. van der put, m. singer

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1、GaloisTheoryofLinearDifferentialEquationsMariusvanderPutDepartmentofMathematicsUniversityofGroningenP.O.Box8009700AVGroningenTheNetherlandsMichaelF.SingerDepartmentofMathematicsNorthCarolinaStateUniversityBox8205Raleigh,N.C.27695-8205USAJuly2002iiPrefaceThisbookisanintroductiontothealgebraic,alg

2、orithmicandanalyticaspectsoftheGaloistheoryofhomogeneouslineardifferentialequations.AlthoughtheGaloistheoryhasitsoriginsinthe19thCenturyandwasputonafirmfootingbyKolchininthemiddleofthe20thCentury,ithasexperiencedaburstofactivityinthelast30years.Inthisbookwepresentmanyoftherecentresultsandnewappro

3、achestothisclassicalfield.Wehaveattemptedtomakethissubjectaccessibletoanyonewithabackgroundinalgebraandanalysisatthelevelofafirstyeargraduatestudent.Ourhopeisthatthisbookwillprepareandenticethereadertodelvefurther.Inthisprefacewewilldescribethecontentsofthisbook.Variousresearchersareresponsiblefo

4、rtheresultsdescribedhere.Wewillnotattempttogiveproperattributionsherebutreferthereadertoeachoftheindividualchaptersforappropriatebibliographicreferences.TheGaloistheoryoflineardifferentialequations(whichweshallrefertosimplyasdifferentialGaloistheory)istheanalogueforlineardifferentialequationsofthe

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

6、racteristiczero.Alineardifferentialequationisanequationoftheform∂Y=AYwhereAisann×nmatrixwithentriesinkalthoughsometimesweshallalsoconsiderscalarlineardifferentialequationsL(y)=∂ny+a∂n−1y+···+ay=0(theseobjectsn−10areingeneralequivalent,asweshowinChapter2).Onehasthenotionofa“splittingfield”,thePicar

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