微分系统的Darboux与Liouville可积张祥上海交通大学数学系，上海200240摘要：Singer于1992年证明了平面多项式微分系统的Darboux可积与Liouville可积的等价性,即平面多项式微分系统有一个Liouville首次积分当且仅当它有一个Darboux积分因子.本文将推广Singer的结果到任意有限维多项式微分系统.证明了如下结论:如果一个n维自治多项式微分系统有n−1个函数独立的Darboux型Jacobi乘子,则该系统有n−1个函数独立的Liouville首次积分(称之为Liouville可积).反之如果该系统有n−1个函数独立的Liouville首次积分,则它必有Darboux型Jacobi乘子.关键词：Darboux可积;Liouville可积;Jacobi乘子.中图分类号：O175.14;O175.12DarbouxandLiouvilleintegrabilityofDifferentialSystemsZHANGXiangDepartmentofMathematics,ShanghaiJiaoTongUniversity,Shanghai200240Abstract:Singerin1992provedtheequivalencebetweenDarbouxianintegrabilityandLiouvillianintegrabilityofplanarpolynomialdifferentialsystems.Thatis,aplanarpolynomialdifferentialsystemshasaLiouvillianfirstintegralifandonlyifithasaDarbouxianintegratingfactor.ThispaperwillextendSinger’sresulttoanyfinitedimensionalpolynomialdifferentialsystems.Themainresultsarethefollowing.Ifanndimensionalautonomouspolynomialdifferentialsystemhasn−1functionallyindependentJacobianmultipliersofDarbouxtype,thenithasn−1functionallyindependentLiouvillianfirstintegrals(calledLiouvillianintegrable).Converselyifthesystemshasn−1functionallyindependentLiouvillianfirstintegral,thenitmusthaveaJacobianmultiplierofDarbouxtype.Keywords:Darbouxintegrability;Liouvilleintegrability;Jacobianmultiplier.基金项目：NationalNaturalScienceFoundationofChina(11271252),RFDPofHigherEducationofChina(20110073110054)作者简介：ZhangXiang(1965-),male,professor,majorresearchdirection：Ordinarydifferentialequationsanddynamicalsystems.Correspondenceauthor：ZhangXiang(1965-),male,professor,majorresearchdirection：Ordinarydifferentialequationsanddynamicalsystems.-2-ithasanintegratingfactoroftheformf1m1...fp0BackgroundandstatementofthemainresultsThetheoryofintegrabilityfordifferentialsystemsisclassic[1]anditisusefulinthestudyofdynamicsofdifferentialsystem.Integrabilityhasdifferentdefinitionsindifferentfields.Herewemainlyconcernthealgebraicaspectsofintegrabilityforpolynomialdifferentialsystems,whichinvolvesanalysis,algebraicgeometry,thefieldextensionandsoon.Forfurtherinformationonthissubject,wereferreaderstoDaboux[2,3],Jouanolou[4],PrelleandSinger[5],Singer[6],Schlomiuk[7],Llibre[8],DumortierandLlibreetal[9],Christopheretal[10,11]andLlibreandZhang[12,13,14,15].DarbouxtheoryofintegrabilitywasestablishedbyDarboux[2,3]in1878forpolynomialdifferentialsystemsofdegreenbyusingtheinvariantalgebraiccurves(resp.surfacesorhypersurfaces)indimension2(resp.3orn>3).Jouanolou[4]in1979extendedtheDarboux’stheorytoconstructrationalfirstintegralswiththehelpofalgebraicgeometry.AnelementaryproofofJouanolou’sresultwasprovidedrespectivelybyChristopherandLlibre[16]in2000fortwodimensionalsystemsandbyLlibreandZhang[13]in2010foranyfinitedimensionalsystems.OnfurtherextensionstoDarbouxtheoryofintegrability,Christopher,LlibreandPereira[11]in2007tookintoaccountnotonlythenumberofinvariantalgebraiccurvesbutalsotheirmultiplicitiesfortwodimensionaldifferentialsystems.LlibreandZhang[12]furtherextendedChristopheretal’sresultin[11]toanyfinitedimensionaldifferentialsystems,wheretherearesomedeepcharacterizationsonthenumberofexponentialfactorsandthemultiplierofinvariantalgebraichypersurfa...