References1BBuchbergerMKauersGroebnerbasisScholarpedia5107763October2010httpwwwscholarpediaorgarticleGroebnerbasis2GCarraFerroAsurveyondi erentialGrobnerbasesinGrobnerBas ID: 365127
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gebrasystemscanbeprogrammedtodo\completion"fastandeciently.Whenappliedtopolynomials,aninvolutivebasisisaGrobnerbasis[1],sonamedbyBuchbergerin1960inhonorofhissupervisor.Allmajorcomputeralgebrasys-temsuseavariantofBuchberger'sfamousalgorithmtocomputeGrobnerbases.MattersarefarmorecomplicatedforsystemsofPDEsandmany\di erentialgeneralizations"ofBuchberger'salgorithmarecurrentlyavailable.See,e.g.,[3]forareviewofearlydi erentialGrobnerbasisalgorithmsandtheirapplicationtosymmetryanalysis,and[2]forarecentsurveyofdi erentialGrobnerbases.Obviously,thetheoryofinvolutivebaseslargelyparallelsthetheoryofGrobnerbasesandSeiler'sbookdrawsheavilyonthatparallelism.Thebookhastenchapters(covering500pages)andthreelongappendices(another100pages).Chapter1givesashortoverviewofthetypeofproblemsthatwillbetreatedinthebook.InChapter2,Seilerintroducesjetbundlesfromtwopointsofview:a\pedestrian"approachbasedonTaylorseriesinlocalcoordinates,andacoordinate-independentapproachthatstressestheintrinsicpropertiesofjetbundles,prolongations,andprojections.Withjetbundlesathand,Seilercontinueswithageometricdenitionofsystemsofdi erentialequationsandonlyresortsto(local)coordinatesinexplicitcomputations.Nodoubt,onehastoresorttoalgebraicmethodstogettotheheartofinvo-lution.Chapter3introducestheconceptofinvolution{themainthemeofthebook{inapurelyalgebraicframework,whichatrstsightisnotatallrelatedtodi erentialequations.Asaninterlude,Chapter4reviewsconcretealgorithmsforthecomputationofinvolutivebases(heavilyinspiredbyworkofGerdt,Blinkov,andZharkov).Inparticular,Seilerdescribestheoptimizedalgorithmthatunderliesmostimplemen-tationsofinvolutivebasesincomputeralgebrasystems.Chapter5probesdeeperintothetheoryofinvolutivebases.EmphasisisonthepropertiesofPommaretbaseswhichappearinsubsequentchapterswhereSeilergivesaconstructivedenitionofinvolutionfordi erentialequations.ReadersinterestedinspecialpropertiesofPommaretbases,suchas�regularity,Reesdecomposition,andsyzygytheory,willndwhattheyneed.StartingwithSpencercohomologyandthedualKoszulhomology,Chapter6discussesthehomologicalinterpretationofPommaretbases.InChapter7,Seilerreturnstodi erentialequationsandappliesthealgebraictheorytotheanalysisofsymbols,whichallowshimtogivearigorousdenitionofunder-andoverdeterminedequations.Thatchapteralsoaddressesthefundamen-talquestion\howdoesonebringanarbitrarydi erentialequationintoinvolutiveform?"Chapter8isdevotedtodetermininganabstractmeasureforthesizeoftheformalsolutionspace.Applicationsincludethecomputationofdi erentialrelations(viz.,Backlundtransformations)betweentwodi erentialequations,andremovingthee ectofgaugesymmetries.Chapter9dealswiththeexistenceanduniquenessofsolutionsofdi erentialequations.ThechallengeistoextendtheCauchy-KovalevskayaTheorem(applica-bletoanalyticnormalsystems)toinvolutivesystems.Thatsubjectisnotnewandearlye ortsresultedintheoriesduetoRiquierandJanetandCartanandKahler,thelatteruseddi erentialformsinsteadofthejetbundleformalism.Seilerconsid-ersvariousalternativesfromthevastliteratureon\existenceanduniqueness"ofsolutionsofnormalsystems.Chapter9alsohasanovelpresentationofVessiot's2 References[1]B.Buchberger,M.Kauers,Groebnerbasis,Scholarpedia,5(10):7763,October2010;http://www.scholarpedia.org/article/Groebner_basis.[2]G.CarraFerro,Asurveyondi erentialGrobnerbases,inGrobnerBasesinSymbolicAnalysis,M.RosenkranzandD.Wang,eds.,RadonSeriesonComputationandAppliedMathematics,vol.2,WalterdeGruyter,Berlin,2007,pp.43-73.[3]W.Hereman,SymbolicsoftwareforLiesymmetryanalysis,inCRCHand-bookofLieGroupAnalysisofDi erentialEquations,vol.3:NewTrendsinTheoreticalDevelopmentsandComputationalMethods,N.H.Ibragimov,ed.,CRCPress,BocaRaton,Florida,1996,pp.367-413.WillyHeremanColoradoSchoolofMines4