POSITIVE POLYNOMIALS ON PROJECTIVE LIMITS OF REAL

POSITIVEPOLYNOMIALSONPROJECTIVELIMITSOFREALALGEBRAICVARIETIESSALMAKUHLMANNANDMIHAIPUTINARAbstract.Werevealsomekeygeomtericaspectsrelatedtonon-convexoptimizationofsparsepolynomials.Themainresult,aPositivstellensatzonthefibreproductofrealalgebraic,affineva-rieties,isiteratedtoacomprehensiveclassofprojectivelimitsofsuchvarieties.Thisframeworkincludesasnecessaryingredientsrecentworksonthemultivariatemomentproblem,disintegrationandprojectivelimitsofprobabilitymeasuresandbasictechniquesofthetheoryoflocallyconvexvectorspaces.Avarietyofapplica-tionsillustratetheversatilityofthisnovelgeometricapproachtopolynomialoptimization.1.IntroductionTheubiquousdualitybetweenidealsandalgebraicvarietiesisre-placedinsemi-algebraicgeometrybyadualitybetweenpre-orders,orquadraticmodulesinaring(seethepreliminariesfortheexactdefini-tions)andtheirpositivitysets.Thisisalreadyanon-trivialdeparturefromclassicalalgebraicgeometry,wellstudiedandunderstoodonlyinthelastdecadeswithtoolsfromrealalgebra,logicandfunctionalanalysis,see[26]forarecent,updatedintroduction.Significantappli-cationstopolynomialoptimizationhaverecentlyemergedfromsuchDate:April4,2007.1991MathematicsSubjectClassification.14P10,28A50,46A22,65K10.Keywordsandphrases.Sparsepolynomial,non-convexoptimization,momentproblem,fibreproduct,projectivelimit.Thisresearchwasstartedat,andsupportedinpartbytheInstituteforMathe-maticsanditsApplicationswithfundsprovidedbytheNationalScienceFounda-tion.ThefirstauthorwassupportedbyanNSERCDiscoveryGrant,CanadaandthesecondauthorbytheNationalScienceFoundation-USA.12KUHLMANNANDPUTINARabstractstudiesinrealalgebraicgeometry,seeforinstancethesurvey[10].Inspiredbysomerecent,farreachingresultsintheoptimizationofpolynomialswithasparsepatternintheircoefficients,advocatedbyKojima,Lasserreandtheircollaborators,weproposeinthisarticleageneralframeworkforStriktpositivstellens¨atzeonconvexconeswhicharemoregeneralthanthewellstudiedpreordersorquadraticmodules.Ourapproachisbasedonthefollowinggeometricconstruction:Letfi:Xi−→Y,i=1,2,betwomorphismsofrealalgebraic,affinevarieties,andletX1×YX2bethe(reduced)fibreproduct,withprojectionmapsui:X1×YX2−→Xi.GivenconvexconesCi⊂R[Xi]withpositivitysetsK(Ci)⊂Xi,weprovidealgebraiccertificatesforelementsofu∗1R[X1]+u∗2R[X2]tobepositiveonu−11K(C1)∩u−12K(C2).Byiteratingthisconstructionoveracertainclassoforientedgraphsweincorporateinourgeometricframeworkthemainresultscirculatinginoptimizationtheory[15,23,20,21,33]andprovidenewapplications,forinstancetoglobaloptimizationonunboundedsets.Thepresentpaperunifiesandextendsinanaturalwaytherecentresultsofthefirstauthor[17,18]andsomeolderobservationsandmethodsduetoSchm¨udgen[31]andthesecondauthor[27].Ourproofsuseaseparationofconvexsetsbylinearfunctionals.Andwhendealingwithfunctionalswhicharenon-negativeonconvexconesofsquaresofpolynomials,representingthembypositivemea-suresisamostdesiredobjective.Thus,viathisavenue,weareledtodisintegrationphenomenaandexistenceofprojectivelimitsofproba-bilitymeasures.Theexsitentresults(someofthemclassical,suchasKolmogorov-Bochner-Prokhorovtheoremontheexistenceofprojectivelimitsofprobabilitymeasures)playakey,complementaryroletoourgeometricstudy.Inthisway,inparticular,wecantreatwiththesametechniquesholomorphicfunctionsofaninfinityofvariables,orpositivityofpoly-nomialsonnon-semi-algebraicsets.POSITIVEPOLYNOMIALSONPROJECTIVELIMITS3Thelastpartcontainsanabundanceofexamples,someofthemdeal-ingwithclassicalfibreproducts,otherswithunboundedsupportingsetsofpositivity,otherswithgroupactionsortrigonometricpolynomials.Wehavenottouchedinthisarticle,butplantodoitinafutureone,thenaturalandverypossibleextensiontohermitiansumsofsquaresandseveralcomplexvariables.Theratherlengthy,inhomogeneousbutnecessarypreliminariesmakedifficultalinearreadingofthisarticle.Weproposethereadertostartwiththesectioncontainingthemainresultsandfilltheneededinfor-mationwithpreliminariesintheway.However,amongthelatter,therearesomeresultsofindependentinterest,suchasTheorem3.1,wichcanberegardedasatruncatedversionofaclassicaltheoremofHavilandandasageneralizationofatheoremofTchakaloff,seeforinstance[29]fordetails.2.Preliminaries2.1.QuadraticmodulesandPositivestellens¨atze.LetAbeacommutativeringwith1.ForsimplicityweassumethatQ⊂A.AquadraticmoduleQ⊂AisasubsetofAsuchthatQ+Q⊂Q,1∈Qanda2Q⊂Aforalla∈A.WedenotebyQ(M;A)thequadraticmodulegeneratedinAbythesetM.ThatisQ(M;A)isthesmall-estsubsetofAwhichisclosedunderadditionandmultiplicationbysquaresa2,a∈A,containingMandtheunit1∈A.IfMisfinite,wesaythatthequadraticmoduleisfinitelygenerated.Aquadraticmodulewhichisalsoclosedundermultiplicationiscalledaquadraticpreordering.Intheterminologyusedthroughoutthisnote,arealalgebraic,affinevarietyX⊂Rdisthecommonzerosetofafinitesetofpolyno-mials,andthealgebraofregularfunctionsonXisA=R[X]=R[x1,...,xd]/I(X),whereI(X)istheradicalidealassociatedtoX.Thenon-negativitysetofasubsetS⊂R[X]isK(S)={x∈X;f(x)≥0,f∈S}.Thedualitybetweenfinitelygeneratedquadraticmodulesandnon-negativitysetsplaysasimilarroleinsemi-algebraicgeometrytotheclassicalpairingbetweenidealsandalgebraicvarieties,se