Butallthesefeaturesarealreadysatisfiedbyarandomwalkwhosejumpsare“small”(forinstance,of finitevariance).Indeed,recallthatarandomwalkisaprocess S = {St : t ∈ N ∪{0}} suchthat S0 =0 (thisistakenarbitrarily)and St = t j =1 Xj , t ≥ 1, where {Xj : j ∈ N} isasequenceofi.i.d.randomvariables,whichweshallassume,withoutanylossof generality,thattheyarecenteredandwithvariance0 <σ2 < ∞.Sowhywouldweneedtogiveitanother
2.Thesecondone,whichisabitmoreinteresting,isthatthedisplacementsoftheprocessareindependentandstationary;thatis,if0 ≤ s < t ≤ s ′ < t ′,thentherandomvariables St Ss and St′ Ss ′ areindependentandthelawof St Ss dependsonlyon t s.Indeed,fortheindependenceofthe increments,onehasthat
Sincetherandomvariables Xj aremutuallyindependent,weconcludethattheincrementsareindependent.Asforthesecondpart,thefactthatthelawoftheincrement St Ss dependsonlyon t s means,inoursetting,thatthelawdependsonlyonthenumberofvariables Xj involved.Sincethey areindependentandidenticallydistributed,theclaimfollows.
3.Thelastproperty,butnotlessimportant,isthat,duetotheCentralLimitTheorem,for t ≥ 0large enough,androughlyspeaking, St ∼N (0, σ 2t).
Inotherwords,thelongtermbehaviouroftherandomvariable St isdescribedbyacenteredGaussian randomvariablewithvariance σ2t.Sinceitdependslinearlywithtime,onecansaythattheprocess isdiffusiveinthelongterm.
Theproofofthisresult(whichisaresultofconvergenceofprobabilitymeasures),relies,mainly,on Prohorov’sTheorem,whichgivesacharacterizationofthefamilyoflawsinducedbythefamilyofrandom functions {Y (n) : n ∈ N},with Y (n) = {Y (n) t : t ∈ [0,1]} intermsofthetopologicalpropertiesofthe space C [0,1],andthefactthatthefinitedimensionaldistributionsofacontinuousstochasticprocess determineitslaw(wereferto[1,Theorems5.1,5.2andp.84]foraproofoftheseclaims).Aproofof Theorem 2.2 foraparticularcaseofrandomwalkisgivenin[2],andageneralproofcanbefoundin[1, Section8]aswell.
dXt = f (t, Xt ) dt, t ≥ 0, (2) modelingsomephenomenawhichweareinterestedinandwhere f : R+ × R → R issomegoodenough function.
Insomecases,thedescriptiongivenbytheODEmightbeabittoosimpleormightnottakeinto accountsomefactorswhichmighthavebeenneglectedduetoasimplificationorduetothefactthat wecannoteasilycontrolthem.Tosolvethis,onecandiscretizetheODEandaddarandomperturbation whichmightevolvewithtime,say V = {Vt : t ∈ R+},leadingto
where g : R+ × R → R issomefunctionmodellingtheintensityoftherandomperturbation.Usually, thisintroducednoiseaccountsforthesuperpositionofseveralsmall(offinitevariance)factorswhich cannotbecontrolled.Hence,andduetotheCentralLimitTheorem,wecanassumethatthelawofthe increments∆Vt isnormallydistributedwithvanishingmean(sincethemeantrajectoriesshouldcoincide withtheonemodeledby(2))andwithvariance∆t.Thelineardependenceontimeinthevarianceis chosenbecause,inmostscenarios,theobservedperturbationcanbesaidtobediffusive.
Sotheonlythinglefttodoistogiveameaningtoexpressions t 0 Xs dBs (stochasticintegral)forasuitable classofstochasticprocesses X = {Xt : t ∈ R+} tosolvetheproblem.
where0= t0 < ··· < tn = T isapartitionofafinitetimeinterval[0, T ]andwhere t ∗ j ∈ [tj , tj +1), j =0,..., n 1.Ideally,onewouldwanttheabovesumstoconvergetothesamelimit(thislimitmight beinprobabilityorinmeansquare,forinstance)nomatterwhatchoiceof t ∗ j ismade.Unfortunately,this isnotthecase,leadingtodifferentdefinitionsofthestochasticintegraldependingonthechoiceofthe midpoints t ∗ j , j =0,..., n 1.Inthiswork,wewillbeconsideringtheleftendpointapproximations(t ∗ j = tj ), whichleadtotheItˆointegral.
ArandomwalkapproachtoStochasticCalculus
Asonemightexpect,thisintegralwillnotbedefinedforanyprocess X .Returningtothediscretization oftheSDE,wehavethattheinformationwehaveontheprocess X attime t +∆t canbedeterminedby theinformationwehaveon Xt andtheinformationwehaveonthedrivingnoise(inourcase,theBrownian motion)attime t +∆t.Atthesametime,theinformationwehaveof Xt dependsontheinformationone hason Xt ∆t andsoon.Allinall,weseethatwecaninfertheinformationof Xt attime t byknowing theentireinformationofthedrivingprocess B untilthattime.Inparticular,theinformationwehaveon Xt doesnotdependontheinformationwehaveattime s for s > t,sotheprocess X cannotseeintothe future.Inthiscase,wesaythattheprocess X mustbeadaptedtothefiltrationgeneratedbythedriving noise(theinformationwehaveon Xt dependsonthehistoryofthenoiseuntilthattime).
Anothernaturalhypothesisontheprocess X isthatitmustbeintegrableinsomesensesothatwe cantalkaboutitsintegral.Moreprecisely,wewillrequirethat
Forthisintegraltobewelldefined,wewillrequire,aswell,theprocess X ,thoughtasamap X :Ω×[0, T ] → R,(ω, t) → X (ω, t)= Xt (ω),whereΩisthesamplespace,tobejointlymeasurablewithrespecttothe corresponding σ-fields.
Whenallthesehypothesisarefulfilled,onecanshowthatintegralslike t 0 Xs dBs canbedefinedasan L2(Ω)-limit(meansquarelimit)ofRiemann–Stieltjessums.Toshowthis,andasitiscustomaryinthis typeofconstructions,onefirstdefinesaclassofsimplefunctionsoftheform
where {ej : j =0,..., n 1} areboundedrandomvariablessuchthattheinformationwehaveon ej depends onlyonthehistoryoftheBrownianmotion(thedrivingnoise)untiltime tj and0= t0 < ··· < tn = T Forsuchfunctions,theintegralwithrespecttotheBrownianmotionisdefinedasthesum(4),where Xt∗ j mustbereplacedby ej
Next,onechecksthat ||·|| definesanormonthespaceofprocesses X satisfyingthepreviously mentionedhypothesisandthatsuchnormedspace(fromnowon,thespaceofItˆointegrableprocesses)is complete.
Finally,oneshowsthatanyprocess X inthenormedspacecanbeapproximatedbysimplefunctions(5), whichallowsustodefinetheintegral T 0 Xs dBs asan L2(Ω)-limitofintegralsofsimpleprocesses.Tojustify thislaststep,acrucialresultforstepfunctions(whichalsoholdsforgeneralItˆointegrableprocesses X )is needed.Weshallstatetheresult,asitwillbeusefulinthefutureforotherpurposes.
Anotherimportantfeatureofthisintegralisthat,when X isadeterministicItˆointegrableprocess(that is,themap X :Ω × [0, T ]isconstantinthefirstargument),onehasthattheprocess I = {It : t ∈ [0, T ]} definedby It = t 0 Xs dBs isaGaussianprocess.Moreprecisely,
Theorem3.2. IfX = f = {ft : t ∈ [0, T ]} isadeterministicItˆointegrableprocess,thenIisacentered Gaussianprocesswithindependentincrementssuchthat,foreach 0 ≤ s < t ≤ T,
t Is =
s fs dBs ∼N 0, t s f 2 u du
Thatis,theincrementisnormallydistributedwithzeromeanandvariance t s f 2 u du.
Thefirstthingonemustcheckwhenonestudiesequationslike(3)(oritsdifferentialform),istomakesure thatthereisatleastonesolutionand,ifpossible,toseethatitisunique.Itturnsoutthat,undersimilar hypothesistotheonesusedinPicard’sTheoremontheprocesses g and f ,onecanshowthatthereisa uniquestochasticprocess X satisfyingequation(3).However,theuniquenessisunderstoodinthesense thatanyotherprocesssatisfyingtheSDEisamodificationofoursolution.
Moreparticularly,werequire f and g tobeLipschitzfunctionsandoflineargrowthwithrespecttothe secondvariableforeach t ∈ [0, T ]:
forsomepositiveconstants C and D,andtheinitialcondition X0 tobedeterministic(thislasthypothesiscan berelaxedbyconsideringanysquareintegrableinitialconditionsatisfyingsomemeasurabilityproperties). Foraprecisestatementoftheresultandaproof,wereferto[4,Theorem5.2.1].
Withallthis,wecanfinallystudyequationsliketheLangevinone,equation(1),whenthenoise Ft is identifiedwiththedifferentialoftheBrownianmotion.Inthefollowingsectionwetreataparticularcase ofsuchequationsandcomputesomeobservablequantities.
forsomepositiveconstant µ andsomerealconstant σ.Thetheoremofexistenceanduniquenessof solutionstellsusthat,foreach T ≥ 0andanydeterministicinitialcondition X0,thereisauniqueprocess (modulomodifications) X = {Xt : t ∈ [0, T ]} satisfyingtheaboveequation.Togiveanexplicitformula for Xt ,wemultiplytheSDEbytheintegratingfactor e µt ,whichleadsto
Theusualproductrulewouldtellusthattheleft-handsidecanbeidentifiedwith d (e µt Xt ).However, thismightnotbetrueinthecontextofstochasticprocesses.Tomakesurethatthisholds,weapplyItˆo’s formulatothefunction F (t, x)= xe µt ,forwhichwehave
So,indeed,wehavethat
or,inintegralform,
Whichsimplifiesto
Withthisandotherresultsliketheisometryformula,wecancomputesomeobservablequantitieslikethe mean,thevarianceandthecovariance.Astraightforwardcomputationusingtheisometryformulashows that
Moreover,Theorem 3.2 tellsusthattheprocess X isGaussianwithmeanandcovariancefunctionsgiven bythefirsttermin(6)and(7),respectively,andthat,foreach t ∈ [0, T ], Xt isanormalrandomvariable withmeanandvariancegivenbythefirstandlasttermsin(6).
forsomegivenfunction K thatblowsupattheorigin.Singularintegralsshowupinanumberofproblems ofanalyticnature.Forinstance,theygeneratesolutionsofsomepartialdifferentialequations,theyarise incomplexanalysis,theyunderpinapparentlyunrelatedsettingsingeometricmeasuretheory,etc.See Figure 1 foranillustrativeexample.
Figure1:AppearanceoftheHilberttransform(themosticonicexampleofsingularintegralin R)in Dirichlet’sproblemfortheLaplaceequation.First,let f bedefinedontheaxis y =0.Obtain u such that∆u =0intheupperhalfplaneand f istheboundaryvalueof u.Then,gettheconjugateharmonic function v of u (theonethatturns u(x, y )+ iv (x, y )intoaholomorphicfunctiononthecomplexplane). Finally,obtaintheHilberttransformof f , Hf ,bycomputingthelimitlim y →0 v (x, y ).
Wearegoingtorunanalgorithminordertoconstruct F andΩ.SetΩ= ∅ andthestep s =1.We spliteachofthecubes Q 0 k into2n dyadicdescendantcubesofthesamesize Q 1 k
Case1: Foreachdescendantcubeinstep s (thatis,foreach k ∈ Z),if
then Q s k isselectedtotakepartinthesetΩ,soupdateΩnew =Ωold ∪ Q s k .Forsuchacube Q s k ,assume that Q s 1 r isitsdirectancestor.Then,by(2)andthefactthat Q s 1 r fellintoCase 2,
whichproves(1)for Q s k
Case2: Instead,if
thenweiterateandfurtherdivide Q s k into2n identicaldescendantcubes(eachwithhalfthesidelengthof theancestor),andcheckintowhichofthetwocaseseachofthemfalls.
ExtensionsoftheCalder´on–Zygmundtheory
Update s new = s old +1andletthealgorithmrunrecursively.Thisway,weobtainthedesiredpartition Rn = F ⊔ Ω,ΩbeingtheunionofallthosecubesthatfellintoCase 1,and F beingthecomplement ofΩ.Plus,(b)hasbeenverifiedforallcubes Q s k thatwereselectedforCase 1.Fact(a)followsfromthe Lebesguedifferentiationtheorem:if x ∈ F ,thismeansthatthereexistsasequenceofnesteddyadiccubes containing x,(Q s k (s))s∈N,(Q s k (s)) ⊃ (Q s+1 k (s+1))beingdirectdyadicdescendants ∀ s ∈ N,suchthatallof thesecubesfellintoCase 2,implyingthat
Thisdecompositionofthedomain Rn of f leadstoausefuldecompositionofthefunction f itself.By defining
(x):=
and b(x):= f (x) g (x),wereachthefollowingcorollary.
Corollary2.2 (See[5,Chapter2,Theorem1]). Letf ∈ L1(Rn ) and λ> 0.Thereexistsadecomposition offassumoftwofunctions,f = g + bsuchthat:
(a)g (x) ≤ 2n λ a.e.x ∈ Rn , (b) 1 |Qk | Qk b(x) dx =0 ∀ k ∈ N, (c) 1 |Qk | Qk |b(x)| dx ≤ 2n λ ∀ k ∈ N,
(d) supp(b)= k ∈N Qk and (e)b ≤ fa.e.
Thefunctions g and b areusuallyreferredtoasthe“good”andthe“bad”partof f .Corollary 2.2 is thekeyingredienttoproveTheorem 2.4,thatallowsustoboundsingularintegraloperators.However,as onemayguess,wefirstneedtomakesomeassumptionontheregularityofthesingularkernelfunction. Theminimalknownhypothesisthatsucceedsistheso-calledH¨ormander’scondition.
Definition2.3. Aconvolutionkernel K on Rn issaidtosatisfy H¨ormander’scondition if
Sincetheintegraliscomputedovertheregion {x ∈ Rn : |x| > 2|y |},thesingularityofthekernelis avoidedbothfor x y andfor x.Insomesense,weareaskingthattheglobalvariationofthekernelisnot sowildthatisnotintegrable.Nevertheless,H¨ormander’sconditionisusuallyseenasaweakenedversion ofthestrongercondition |∇K (x)|≤ C |x|n+1 , forall x ∈ Rn awayfromtheorigin.Allinall,hereisthetheoremthatgivesmeaningtothetheory.Inthe literature,onecanfindmanyvariationsandconsequencesofit.
BernatRamisVich
Theorem2.4 (See[5,Chapter2,Sections2and3]). LetTbealinearoperatorsuchthatthereexistsa measurablekernelfunctionKsuchthat Tf (x)= Rn K (x y )f (y ) dy convergesabsolutelywheneverf ∈ L2(Rn ) andx / ∈ supp(f ).Supposethefollowing:
Thestrategyfortheproofis,accountingfortheboundednessassumptionontheHilbertspace L2(Rn ), usingtheCalder´on–Zygmundlemmatofirstshow(b),i.e.,that T isweak-type(1,1).Afterthat,one canusetheMarcinkiewiczinterpolationtheorembetween p =1and p =2toget(a)for1 < p ≤ 2. Eventually,adualityargumentcoversthedualrange2 ≤ p < ∞.
Definition3.2. Let((X , d ),Σ, µ)beameasuremetricspaceandlet f ∈ L1 loc(X )bealocallyintegrable function.The centredHardy–Littlewoodmaximalfunction of f isdefinedas
Similarly,the uncentredHardy–Littlewoodmaximalfunction of f readsas
wherethesupremumistakenoverallballs B containing x Whenthedoublingpropertyisinforce,thenthecentredanduncentredversionareeasilychecked tobecomparable.ItisalsoremarkabletonotethattheHardy–Littlewoodmaximalfunctiondefinesa boundedoperatoron Lp spaces,1 < p < ∞ ([5,Chapter1,Theorem1]).Infact,inordertoshow Lp -boundednessforabroadclassofso-calledCalder´on–Zygmundoperators(thoseunderthehypotheses ofTheorem 2.4 orTheorem 3.5),onecanfirstshow,aspointedout,thattheHardy–Littlewoodmaximal functionis Lp -bounded,andthenusethisspecificresulttoprove Lp -boundednessforthebroadclassof Calder´on–Zygmundoperators.
Lemma3.3 (Calder´on–Zygmundlemmainthegeneralsetting;see[7,Chapter1,Theorem2]). Letf ∈ L1(X ) and λ> 0.ThereexistsapartitionofthespaceX = F ⊔ Ω,Fbeingaclosedsetand Ω anopen set,suchthat
andsoΩ:= {x ∈ X : Mf (x) > λ},beingrespectivelyclosedandopen,because Mf (x)isacontinuousfunctionof x BytheLebesguedifferentiationtheorem,fora.e. x ∈ F , λ ≥ Mf (x)=sup r >0 1
so(a)isshown.
Letusintroducesomenotation.Foraball B = Br (x)centredat x withradius r andforsomeuniversal constants0 < C ∗ < C ∗∗,denoteby B ∗ := BC ∗r (x)and B ∗∗ := BC ∗∗r (x)thecentreddilationsby factors C ∗ and C ∗∗,respectively.Inordertoprove(b),weuseaVitali-typecoveringlemma([7,Chapter1, Lemma2]):giventheclosedset F ,thereexistsasequenceofballs(Bk )k ∈N andtwofamiliesofeach dilations(oruniversaldilationconstants0 < C ∗ < C ∗∗ ),(B ∗ k )k ∈N and(B ∗∗ k )k ∈N,suchthat (a)(Bk )k ∈N arepairwisedisjoint, (b) k B ∗ k = F c ,and (c) B ∗∗ k ∩ F = ∅, ∀ k
BernatRamisVich
Itisconvenienttoextractanothersequenceofsets.Takethefirstelementin(B ∗ k )k ∈N anddefine Q1 := B ∗ 1 Next,define Q2 := B ∗ 2 ∖ (Q1).Byaninductiveprocess,build
Itisdirectlydeducedthatthesets Qk satisfy k Qk = F c justlikethe B ∗ k ,althoughwiththeadvantage thatthe Qk arepairwisedisjoint.Thedownside,comparedtothe B ∗ k ,isthatthe Qk arenolongerballs, butotherlesselementarysets.Thename Qk ofsuchnewsetsisinspiredbytheirroleintheproofof Theorem 3.5,whichmimicstheonecarriedoutbythecubesintheproofofthe X = Rn case.
Now,foreach Bk inthesequence(
N,chooseapoint
.Bythedefinitionof F ,
where C dp istheconstantfromthedoublingproperty(seeDefinition 3.1)and C unc istheconstantinthe equivalence
Infact, C unc =(C dp) 1.Thetwolastinequalitiesin(5)stemfromthefactthat B
⊆ Qk ⊆ B ∗∗ k andthedoublingproperty:
partitionΩ, Ω= k Ωk ≡ k Qk ,theproofiscomplete.
Notethatthisproofunveilsthepreciseidentityofthesets F andΩ,whicharedefinedintermsofthe Hardy–Littlewoodmaximalfunction.
InexactlythesamewayasinCorollary 2.2,theCalder´on–Zygmunddecompositionofanintegrable function f ∈ L1(X )as f = g + b isdeduced.
Wementionedthatwewishourgeneralisedtheoremtoholdforvector-valuedfunctions.Theconstructionofthe Lp spacesforsuchfunctionsisnowadaysstandard([4,Chapter5]).Letusdenoteby Lp B (X ) theLebesguespaceof Lp -integrablefunctionsonsomemeasurespace X andtakingvaluesintheBanach space B.Thisis,for1 ≤ p < ∞,set
p B (X ):= F : X → B : X
F (x)
p B d µ(x) < ∞ , whereasfor p = ∞,
Additionally,denoteby L(A, B)theBanachspaceofalllinearandcontinuousmapsbetweenBanach spaces A and B
Definition3.4. Let A and B beBanachspaces.Anoperatorkernel K ontheproductmeasurespace ((X , d ),Σ, µ) × ((X , d ),Σ, µ)takingvaluesin L(A, B)issaidtosatisfy H¨ormander’scondition if
forsomeconstant C > 1.
Anotherimportantremarkisthatnow,thekerneloperatorinvolvestwoentriesinsteadofjustone, comparedtotheconvolutionoperators.Thereasonforthisisthat“x y ”doesnotmakesenseingeneral measuremetricspaces,sincetheylackthevectorspacestructure.Thus,wegetaroundthisissueby inputtingtwovariables x ∈ X and y ∈ X ,withtheunderstandingthatthekernelissingulararound x = y Astonishingly,thenaturalgeneralisationofTheorem 2.4 turnsouttoworkinthissettingaswell!
Theorem3.5 (See[7,Chapter1,Theorem3]and[4,Chapter5,Theorem3.4]). Let ((X , d ),Σ, µ) bea measuremetricspacewiththedoublingproperty.LetA,BbeBanachspacesandletTbealinearoperator whichisrepresentedby
(x)= X
, y )
(y )
(y ), wheneverF ∈ L∞ A (X ) withcompactsupportandx / ∈ supp(F ),wherethevector-valuedkernelK ∈L(A, B) ismeasurableinX × Xandlocallyintegrableawayfromthediagonal.Assumethat
Example3.6 (SmoothLittlewood–Paleysquarefunction) Let Pj besmoothLittlewood–Paleyprojectors. Namely,the Pj aredefinedasmultipliersontheFourierside:
BernatRamisVich
Here, ψ isasmoothcompactlysupportedfunction,thedyadicdilationsofwhichformapartitionofunity infrequency.Thisway, Pj f capturesthe“part”of f withfrequenciesaround2j
whichisa linear operatormappingfunctionstosequencesoffunctions.1 Accordingly,
Webroughtthesquarefunctiontothevector-valuedsetting.Atthispoint,onewouldattempttoapply Theorem 3.5 to Sf .Nonetheless,adirectapplicationfailstoshowthat Sf isboundedon Lp (Rn )for1 < p < ∞.ItisnecessarytocombineTheorem 3.5 withaprobabilistictrickinvolvingRademacherrandom variablestoeventuallyshowthat Sf isboundedon Lp (Rn )for1 < p < ∞
Riemannianmanifolds,introducedinthesecondhalfofthe19thcenturybyBernhardRiemann,aremanifoldsendowedwithascalarproductoneachtangentspace.Usually,oneworkswithan n-dimensional smoothmanifold M andaRiemannianmetric gx : Tx M × Tx M → R,denotingascalarproductthat variessmoothlywith x ∈ M.Thisscalarproductgivesrisetoanormontangentvectors,bysetting ∥v ∥g x = gx (v , v ),andtoalengthforcurves γ :[0,1] → M,bysetting ℓg (γ)= 1 0 ∥γ′(t)∥g γ(t) dt.The scalarproductcanalternativelyberepresentedinalocalchartbyacollectionof n × n positivedefinite andsymmetricmatrices(gij (x ))ij .Thatway,thecanonicalRiemannianmeasuredvg of(M, g )inthislocal chartisgivenbytheformuladvg (x )= det(gij (x ))dx1 ∧···∧ dxn
FinslermanifoldsareageneralisationofRiemannianmanifolds,whereeachtangentspaceisendowed withanorminsteadofwithascalarproduct.Thesemetricstructureswerefirstconsideredin1918byPaul Finsler,althoughtheterm Finslermanifold wascoinedlaterby ´ ElieCartan,in1934.Usuallyanorm ∥·∥ isamapfromavectorspaceto R+ =[0, ∞)thatfulfilsthefollowingconditions: ∥v ∥ =0onlyif v =0, ∥λv ∥ = |λ|∥v ∥ for λ ∈ R and ∥v + v ′∥≤∥v ∥ + ∥v ′∥.InFinslergeometry,non-necessarilysymmetricnorms areconsideredmoregenerallybyreplacingthesecondpropertybythecondition ∥λv ∥ = λ∥v ∥ for λ ∈ R+ ThestructureassociatedtoavaryingnormoneachtangentspaceiscalledaFinslermetricandthenorm atsomepoint x isusuallydenotedby Fx .InanalogytotheRiemanniancase,onedefinesthelengthof acurve γ :[0,1] → M by ℓF (γ)= 1 0 Fγ(t)(γ′(t))dt.However,andincontrasttotheRiemanniancase, thereisnounambiguouslydefinedvolumenotionforFinslermetrics.Twoofthemostusedonesarethe Holmes–ThompsonandtheBusemann–Hausdorffvolumes.Theformerisrelatedtothestandardsymplectic formon T ∗M,and,thelatter,totheHausdorffmeasureofametricspaceinthesymmetriccase.From nowon,only2-dimensionalmanifoldswillbeconsidered.FixinganauxiliaryRiemannianmetric g on M, theHolmes–ThompsonandBusemann–Hausdorffareasaredefinedas
Here, |Bx |g denotestheRiemannianmeasureoftheunitball Bx = {v ∈ Tx M | Fx (v ) ≤ 1},and B ◦ x its polarconvexbodywithrespectto gx .NotethataFinslermetric F isuniquelydefinedspecifyingtheunit spheres Ux = {v ∈ Tx M | Fx (v )=1} ateachpoint x ∈ M
Definition1.1. AFinslermetric F on M issaidtobereversibleif Fx (v )= Fx ( v )forall(x , v ) ∈ TM Inotherwords, F issaidtobereversibleifalltheunitballsarecentrallysymmetric.
Sinceascalarproductinducesasymmetricnormoneachtangentspace,Riemannianmetricsarea particularcaseofFinslermetrics.Assketchedin[5,Proposition3.5],thedefinitionsin(1)areindependent ofthechosenauxiliaryRiemannian g ,andaneasyconsequenceoftheBlaschke–Santal´oinequalityisthe following.
Proposition1.2. IfFisareversibleFinslermetriconamanifoldM,then areaBH(M, F ) ≥ areaHT(M, F ) andequalityholdsifandonlyifFcomesfromaRiemannianmetric.
Definition2.1. ThesystoleofaFinslerclosedmanifold(M, F )whichisnotsimplyconnectedisdefined by
sys(M, F ) :=inf{ℓF (γ) | γ isanon-contractibleloopin M}.
OneexpectsthattheareaofaFinslermanifoldforwhichallnon-contractibleloopshavealength uniformlyboundedfrombelowcannotbemadearbitrarilysmall.Thisisdescribedbyaninequalityofthe form
area(M, F ) ≥ C sys 2(M, F ) holdingforsomesetofmetrics F ,where C issomepositiveconstant.Suchaninequalityiscalledan isosystolicinequality andtheconstantmightdependonthesetofmetricsconsidered.Usuallyoneconsiders eitherRiemannianmetrics,reversibleFinslermetricsorallFinslermetrics.Anisosystolicinequalityissaid tobeoptimaliftheconstant C cannotbeimproved.Finally,itissaidthatthereissystolicfreedomifsuch apositiveconstantdoesnotexist.
Thefirstoptimalisosystolicinequalitywasfoundforthe2-torusin1949byCharlesLoewner.Asit isexplainedbyhisstudentPaoMingPuattheendof[6],Loewnerfounditduringthelecturesofa courseonRiemanniangeometryhewasteachingatthetime.HeprovedthatforanyRiemannianmetric g onthe2-torus,area(T2 , g ) ≥ √3 2 sys2(T2 , g ),andthattheconstant √3 2 isoptimal.InspiredbyLoewner’s method,Puprovedin[6]thatfortherealprojectiveplanearea(RP2 , g ) ≥ 2 π sys2(RP2 , g )foranyRiemannianmetric g andthattheconstant 2 π isalsooptimal.ForthecaseofFinslermetricsandthe2-torus,a completesummaryofoptimalisosystolicinequalitiesisdonein[2].Thisarticlegathersallknownoptimal constants,includingtheonesforRiemannian,reversibleFinslerandnot-necessarilyreversibleFinslermetricsforbothHolmes–ThompsonandBusemann–Hausdorffareas.There, T2 isidentifiedwiththequotient oftheEuclideanplane R2 bytheintegergrid Z2.Inthatcase,ametricon T2 isjustametricon R2 compatiblewiththequotientmap,andnon-contractibleloopsin T2 correspondtopathsbetweenpoints in R2 thatdifferbysome z ∈ Z2 \{(0,0)}.Thestrategyfollowedinthearticleistoreducethegeneralcase tothecasewherethemetricisflat,inthesensethattheunitballsin Tx T2 arethesameforall x ∈ T2 Then,theinequalityismostofthetimesaconsequenceofpreviouslyknownresultsinconvexgeometry. See[2]forallthedetails.
Forthecaseof RP2,ananalogousprocedurewouldbetolookforarbitrarilylargeunitballsthatdonot leadtoanarbitrarilysmallvalueforthesystole.Thisisproventobepossiblein[5,Section6],whichleads totheconclusionthatsystolicfreedomalsoholdsinthenon-reversiblecaseforareaBH.Theideabehindthe constructionin[5]istobuildametricinahemisphereof S2 suchthattheequatorcontainsasystolicloop ofsomefixedlength.InordertohaveasmallvalueforareaBH(RP2 , F ),oneneedstohavelargeunitballs ingreatpartofthehemisphereof S2.However,theselargeunitballs(whichleadtoshortdistances)must besuchthatasystolicloopstillliesinsidetheequator.Thisisdonewithunitballsofarbitrarilylargesize L
UnaiLejarzaAlonso
inonedirectionandarbitrarilysmallsize ε intheoppositedirection,asisshowninFigure 3b.Theseballs areallowedtobearbitrarilylargeandtheypreventcurvesthatgotowardsthepolefrombeingtooshort. Notethattheyareconvexsetscontainingtheorigin,sotheycorrespondtosomenon-reversibleFinsler metric.Thefinalstepistomakesuchametriconahemisphereof S2 well-definedandcompatiblewitha metricon RP2.Firstofall,oneneedstohaveawell-definedunitballatthepole:itcannotdependonthe meridianthatapproachesthepoint.ThiscanbeachievedchangingsmoothlytheunitballinFigure 3b to arotationallyinvariantonearoundthepole.Besides,ametric F on S2 iscompatiblewithametricon RP2 if Fx (u1)= F x (u2),where u1 and u2 arethedifferentliftsofsome v ∈ T[x ]RP2.Geometrically,assume thatoneobserves S2 fromthepointsuchthat x and x aretheclosestandfurthestpointsoftheequator, respectively.AsisillustratedinFigure 2,fromthispointofview, u1 and u2 arehalfturnrotationsofone another.Achangeinpointofviewsothat x isnowinfrontandstillwiththepoleabovecorresponds toahorizontalflipoftheviewof T x S2.Inconclusion,theunitballsofantipodalpointsintheequator mustbeverticallyflippedwhenseenthesamewayasinFigure 3b.Thus,itisenoughtochangesmoothly theunitballsintheequatortoverticallysymmetriconesinordertohaveacompatiblemetric.
L .See[5,Proposition6.2] forthedetails.If γ =(γ1, γ2)doesnotenterinthesmoothenzonearoundthepole,
Notethatequalityholdsif γ′ 2(t)=0and γ1 increasesordecreasesmonotonicallybetweenazimuthal coordinatesthatdifferexactlyin π.If γ entersthesmoothenzonearoundthepole,thefirstpartof γ must jointheinitialpointwiththezone.Bywhathasbeenmentionedabove,thelengthofvectorspointing tothepoleisproportionalto 1 ε .Then,asmallenoughchoiceof ε wouldimplythat ℓF (γ) >π also,and thereforesys(RP2 , F )= π.Intheend,areaBH(RP2 , F )= π 2(ε+L) sys2(RP2 , F ) <
2L sys2(RP2 , F )forany valueof L > 0.Inparticular,since L canbechosenarbitrarilylarge,thereissystolicfreedomfor RP2 and theBusemann–Hausdorffarea.See[5,Section6]formoredetails.
4.Optimalinequalitiesfornon-reversiblemetrics
´ AlvarezPaiva,BalacheffandTzanevprovedin[1,TheoremIV]thatareaHT(T2 , F ) ≥ 3 2π sys2(T2 , F )forflat metricsandthatequalityholdswhentheunitballisthetrianglewithvertices(1,0),(0,1)and( 1, 1). Finally,bythepropertiesofthe stablenorm,onededucesthat areaHT(T2 , F ) ≥ areaHT(T2 , ∥·∥) ≥ 3 2π sys 2(T2 , ∥·∥)= 3 2π sys 2(T2 , F ) alsoforanyFinslermetric.
FindingtheoptimalisosystolicinequalityforthemoregeneralFinslercaseforareaHT and RP2 is stillanopenproblem.Existenceofanoptimalinequalitycanbeprovenbysymmetrisingthemetric. Indeed,consideringthesymmetricmetric ˜ Fx (u)= Fx (u)+ Fx ( u),itcanbeprovenindimension2that
| ˜ B ◦ x |≤ 6|B ◦ x | (see[7,Theorem1]).If γ ⊆ RP2 isasystolicloopfor ˜ F ,theinvertedloop γ isalsononcontractible,andthensys(RP2 , ˜ F )= ℓF (γ)= ℓF (γ)+ ℓF ( γ) ≥ 2sys(RP2 , F ).Joiningtheseinequalities withtheoptimalinequalityforreversiblemetrics,
Conjecture4.1. TheoptimalisosystolicinequalityforFinslermetricsandHolmes–Thompsonareais areaHT(RP2 , F ) ≥ 2 π sys2(RP2 , F )
Theauthorhastriedtoattackthenon-reversiblecaseandHolmes–Thompsonareawithlittlesuccess. Considerthefamilyofmetricsusedintheproofofsystolicfreedomintheprevioussection.Forsimplicity, considerthemetricbeforethesmoothing,whichcanbedoneinanirrelevantarbitrarilysmallregion. Imposingthatthesystoleisstillattainedalongtheequatoramountstoimposingthat 1 ε + 1 L ≥ 2.Indeed, asbefore,if γ doesnottouchthepole, ℓF (γ) ≥ π.Andifittouchesit,itmustgoupandthenback down,havingalength ℓF (γ) ≥ π 2 1 ε + 1 L ≥ π.ThedualconvexbodyoftheunitballsofFigure 3b canbe computedtobetheconvexhullofthepoints(±1,0), 0, 1 ε and 0, 1 L .ThisconvexkitehasLebesgue measure 1 ε + 1 L ,andsimilarlytotheBusemann–Hausdorffcase,by(1), areaHT(RP2 , F )=
sys 2(RP2 , F ).
Inconclusion,areaHT(RP2 , F ) ≥ 2 π sys2(RP2 , F )if 1 ε + 1 L ≥ 2,whichpreventstheexistenceofshortcuts throughthepole.Thesmootheningprocesswouldjustleadtoresultsarbitrarilyclosetotheaboveinequality, agreeingwithConjecture 4.1
Anyunitballcanbedrawninsidearectangleandcontainingatrianglethattouchesthreeofthefurthest pointsfromtheorigin.Thismightleaveshortestlengthsinvariantanditmightbeinterestingtoperform asimilartestfortriangle-shapedunitballs.Forexample,considertriangleswithvertices(1,0),( δ, ε) and( δ, L).Inthiscase,thedualtrianglehasvertices 1 δ ,0 , 1, 1+δ ε and 1, 1+δ L ,andLebesgue measure (1+δ)2 2δ 1 ε + 1 L .Thenormisnotsoeasytocomputebutonecouldexpectthatimposingthatthe systoleisattainedaroundtheequatorwouldimplythesame(orworse)inequality.Itwouldbeasurpriseif thereexistedvaluesfor ε, L and δ thatproveConjecture 4.1 wrong.Theauthor’ssearchofexamplesthat provetheconjecturewronghasbeenunfruitfulandlookingforwaystoproveitmightbemoresensible.
Aminoradvanceinthisdirectionhasbeenachievedin[5,Theorem5.13],givingaslightgeneralisation ofIvanov’sresultforreversiblemetrics.Itstatesthattheinequalityisalsotrueformetricssuchthatthe distancebetweenanytwopointsofasystolicloop γ0 isattainedthrough γ0.Inotherwords,oneneedsto havenoshortcutsbetweenpointsof γ0 thatdeviatefrom γ0.Inthiscase,if γ0 connects x to y (andnot
UnaiLejarzaAlonso
theotherwayaround),thedefinitionofsystoleensuresthattherearenoshortcutsfrom x to y .However, inthenon-reversiblecase,theremightbeshortcutsfrom y to x .Ivanov’sassumptionistohaveareversible metric,whichimpliesthattherearenosuchshortcuts.Theassumptionin[5,Theorem5.13]isweakerbut stillensuresthattherearenosuchshortcuts.TheproofisessentiallythesamethattheoneforIvanov’s theoremalthoughFigure 3a getsslightlymodified.Forinstance,thecurveisnolongercontainedinthe square[0, s]2,andtheshortstraightlinesbecomeunknownbutbounded.Thecorrespondingcurveisshown in[5,Figure6],andtheinequality(3)ismodifiedto
Luckily,forarbitrarilylarge n theinequalitybecomesareaHT(RP2 , F ) ≥
(RP2 , F ).Asufficient conditiontoavoidshortcutsisthatthesystoliccurve γ0 hasthesameforwardandbackwardlength.In particular,thisholdsif Fγ0
Thisshowsthattheaveragingprocedurefailstohavegoodpropertiesevenforthesupremumnorm.As wassuggestedbyF.Balacheff,anotherapproachcouldbetoconsideracontactstructureontheunitary tangentbundle S ∗RP2.Withcontactformsthereisatheoremsimilartotheuniformisationtheoremthat saysthattheinitialcontactformandafixedroundonearecontactomorphic.Onemightbeabletoaverage overthegroupofdiffeomorphismsof S ∗RP2 thatleavestheroundcontactforminvariant.Thisissimilar
tothefactthattheactionofSO(3)leavesaroundmetricon S2 invariant.Itturnsoutthat S ∗RP2 is isomorphictotheLensspace L(4,1).However,thesystoleseemstobemoredifficulttodealwith.
Section 7 studiespyramidalpartitions,whichareanextensionoftriangularpartitionstohigherdimensions.Weprovethatthecharacterizationintermsofconvexhullsgeneralizesnicelyandthat,for dimension3orhigher,thenumberofremovableandaddablecellscanbearbitrarilylarge.Wealsodescribe theresiduemodulo d ofthenumberof d -dimensionalpyramidalpartitionsofsize n,for d prime.
InSection 8,convexandconcavepartitionsareanalyzed.ThesearepartitionswhoseFerrersdiagram canbeseparatedfromitscomplementbyaconvexorconcaveline.Wepresentseveralcharacterizations andwedescribetheirremovableandaddablecellsintermsofconvexhulls.Finally,weprovethatthere existconstants a, b, c suchthatthenumberofconvexpartitionsofsize n isgreaterthanexp(a 3 √n)and smallerthanexp(b 3 √n log n),andthenumberofconcavepartitionsofsize n isgreaterthanexp(c 3 √n).
A partition λ isaweaklydecreasingsequenceofpositiveintegers,calledthe parts of λ.Wewilldenote λ = (λ1, λ2,..., λk ),or λ = λ1λ2 λk whenthereisnopossibilityofconfusion.Wecall |λ| = λ1 + λ2 + + λk the size of λ.If |λ| = n,wesaythat λ isapartitionof n
Let N denotethesetofpositiveintegers.The Ferrersdiagram of λ isthesetoflatticepoints
{(a, b) ∈ N2 | 1 ≤ b ≤ k,1 ≤ a ≤ λb }
Wewilloftenidentifyalatticepoint(a, b)withtheunitsquare(calleda cell )whosenorth-eastcorner is(a, b).Inparticular,wesaythatacellliesabove,beloworonalinewhenthenorth-eastcornerdoes. TheFerrersdiagramcanthenbeinterpretedasasetofcells.Wewilloftenidentify λ withitsFerrers diagram,andusenotationsuchas c =(a, b) ∈ λ
Let σk =(k, k 1,...,2,1)denotethe staircasepartition of k parts.The conjugate λ′ of λ isobtained byreflectingitsFerrersdiagramaboutthe y = x axis.The complement of λ isdefinedtobetheset N2 \ λ, where λ isidentifiedwithitsFerrersdiagram.
Definition2.1. Apartition τ is triangular ifitsFerrersdiagramconsistsofthepointsin N2 thatlieonor belowthelinethatpassesthrough(0, s)and(r ,0)forsome r , s ∈ R>0,calleda cuttingline
Denoteby∆thesetofalltriangularpartitionsandby∆(n)thesetoftriangularpartitionsofsize n Corteeletal.[5]obtainthegeneratingfunctionof |∆(n)| andboundtheasymptoticgrowthofthisnumber.
Theorem2.3 ([5]). Thereexistpositiveconstantscandc ′ suchthat,foralln > 1, cn log n < |∆(n)| < c ′ n log n
Let c =(i , j)beacellofatriangularpartition λ = λ1 λk .Definethe armlength andthe leglength of c tobe a(c)= λj i and ℓ(c)= λ′ i j,thatis,thenumberofcellstotherightof c initsrow,and above c initscolumn,respectively.BergeronandMazin[1]characterizetriangularpartitionsandstudy thenumberofcellsthatcanbeaddedorremovedwhilepreservingtriangularity.
Lemma2.4 ([1,Lemma1.2]). Apartition λ istriangularifandonlyiftλ < t + λ ,where
AsimilarcharacterizationintermsofconvexhullsforasingleremovablecellisprovedbyElizaldeandthe presentauthorin[7],andisthenusedtodescribeanalgorithmthatdetermineswhetherapartition λ of n into k partsistriangular.Saidalgorithmhascomplexity O(k)fortheinitializationand O(min{k, √n}) fortherestofitssteps,whereasanalgorithmbasedonBergeronandMazin’sLemma 2.4 wouldtake time O(n).
https://reportsascm.iec.cat
4.Bijectionstobalancedwordsandefficientgeneration
Inthissection,wepresenttwodifferentinterpretationsoftriangularpartitionsintermsoffiniteSturmian words,alsoknownasbalancedwords.Thefirstinterpretation,whichishintedatin[1],isquitenatural, anditwillallowustoproveanenumerationformulainSection 6.Thesecondonerelateseachtriangular partitiontoabalancedwordtogetherwithtwopositiveintegers,anditwillbeusedinSection 4.4 to implementanefficientalgorithmtocounttriangularpartitionsbysize.
4.1Backgroundonbalancedwords
Afiniteconsecutivesubwordofawordiscalleda factor.Aninfinitebinaryword s is Sturmian if,for every ℓ ≥ 1,thenumberoffactorsof s oflength ℓ isexactly ℓ +1.TheapplicationsofSturmianwords rangefromcombinatoricsandnumbertheorytodynamicalsystems;see[9]forathoroughstudy.
Afinitebinaryword w = w1 wℓ isafactorofsomeSturmianwordifandonlyifitis balanced,that is,foranypositiveintegers h ≤ ℓ and i , j ≤ ℓ h +1,wehave
)|≤ 1.
Thisconditionstatesthatforanytwofactorsof w ofthesamelength,thenumberofonesinthesefactors differsbyatmost1.Denoteby B thesetofallbalancedwords,andby Bℓ thesetofthoseoflength ℓ
Theorem4.5. Themap χ =(min,dif,wrd) isabijectionbetween W andtheset T = {(m, d , w ) ∈ N × N ×B0 | m ≤ d +1; w 1 ∈B0 ifm = d +1}.
Itsinverseisgivenbythemap
(m, d , w1 wk 1)= τ1 τk , where τi = m + k 1 j =i (wj + d ) fori ∈ [k].
Additionally,given τ ∈W withimage χ(τ )=(m, d , w ),itsnumberofpartsequalsthelengthofwplus one,anditssizeis
τ | = km + k 2 d + k 1 i =1 iwi (3)
4.4Efficientgeneration
Beforethiswork,theentryoftheOEIS[10,A352882]forthenumbertriangularpartitionsof n onlyincluded valuesfor n ≤ 39.Thesearethetermslistedin[5],wheretheyareobtainedfromthegeneratingfunction inTheorem 2.2.Thisapproachturnsouttobeimpracticalforlarge n.
Theorem 4.5 canbeusedtoimplementamuchmoreefficientalgorithmthatcanquicklycompute thefirst105 termsofthesequence.Considerthetreewhereeachvertexisabalancedwordoflengthat most ⌊√2N⌋,andtheparentofanonemptywordisthewordobtainedbyremovingitslastletter.On input N,ouralgorithmrunsadepthfirstsearchthroughthistree.
Foreach w ∈Bℓ with ℓ ≤ √2N,thealgorithmfindsallthevalues m, d ∈ N suchthat(m, d , w ) ∈T , asdefinedinTheorem 4.5,andsuchthatthesizefunctiongiveninequation(3)isatmost N.Each triplet(m, d , w )correspondstotwopartitions,thewidetriangularpartition τ = χ(m, d , w )anditsconjugate,exceptwhen w =0k 1 (forsome k ≥ 2)and m = d ,inwhichcaseitaccountsforonlyonepartition, thestaircase σk
Let c and c + betheleftmostandrightmostremovablecellsof τ .Followingthenotationin[1],let τ ◦ bethetriangularpartitionobtainedfrom τ byremovingallthecellsinthesegmentbetween c and c + (or,if c = c +,justremovingthatcell).
Lemma6.1. Forany τ ∈ ∆(n) withn ≥ 1, I (τ )= I (τ \{c })+ I (τ \{c +}) I (τ ◦)+1.
Thisrecurrencerelationcomesfromaninclusion-exclusionargument.Alongwiththebasecase I (ϵ)=1 (where ϵ denotestheemptypartition),itallowsustocompute I (τ )forany τ ∈ ∆,althoughnotvery efficiently.Wewillnowpresentamoreconvenientformulaforthecaseinwhich τ isastaircase.
Weusetheterms height and width ofapartition τ torefertothenumberofpartsandthelargestpart of τ ,respectively.Let∆ℓ×ℓ bethesetoftriangularpartitionswhosewidthandheightareatmost ℓ.Itcanbe provedthatapartitionbelongsto∆ℓ×ℓ ifandonlyifitisatriangularsubpartitionof σℓ.Ournextgoalisto giveaformulafor I (σℓ)= |∆ℓ×ℓ|.Theproofofthefollowinglemmausesthebijection ω fromequation(2).
Lemma6.2. For ℓ ≥ 1,thenumberoftriangularpartitionsofwidthexactly ℓ andheightatmost ℓ is |Bℓ|/2,and
Inthecaseoftriangularpartitionsin N2,wehavethattheonlypartitions τ ∈ ∆suchthat τ = τ ′ (thatis,theyaresymmetricalwithrespecttotheline x = y )arethestaircasepartitions.Fromthisfact, wecandeducethat |∆(n)|≡ 1(mod2)when n = m 2 forsomeinteger m ≥ 2,and |∆(n)|≡ 0(mod2) otherwise.Thisapproachcanbeextendedto d -dimensionalpyramidalpartitionsbystudyinganactionof thesymmetricgrouponthem.Wewilldenoteby∆d D(n)thesetof d -dimensionalpyramidalpartitionsof size n,toavoidconfusionwith∆1(n)and∆2(n)definedinSection 5
Theorem7.4. Letn, d ∈ N,withdaprimenumber.Ifthereexistsanintegerm ≥ dsuchthatn = m d , then |∆d D(n)|≡ 1(mod d ).Otherwise, |∆d D(n)|≡ 0(mod d )
8.Convexandconcavepartitions
ConvexpartitionsaredefinedbyDeanHickersonin[10,A074658],wherethenumberofconvexpartitions ofsize n iscountedfor n ≤ 55.TheconceptofconcavepartitionsisessentialtosomeSchurpositivity conjectures(see[2,Conjecture7.1.1]).Inthissection,wewillextendourresearchontriangularpartitions tothesemoregeneralfamilies,startingwithsomecharacterizations.
Proposition8.4. Acellc =(a, b) isremovablefromaconvexpartition η ifandonlyifitisavertex of Conv(η) and (a +1, b),(a, b +1) / ∈ η.Similarly,acellc ′ isaddabletoaconcavepartition ν ifandonly ifitisavertexof Conv(N2 \ ν)
Toclosethearticle,wewillstudytheasymptoticgrowthofthenumberofconvexorconcavepartitions. Wewilluse (n)(resp. (n))forthesetofconvex(resp.concave)partitionsofsize n
Definition2.2. Givenapoint x inafinitetopologicalspace(X , τ ),wedefinethe minimalopensetofx astheintersectionofallopensetscontaining x :
Theminimalopensetsformabasisforthetopologyof X ,calledthe minimalbasisofX
Definition2.3. A preorder isareflexiveandtransitiverelation.A preorderedset or preset isasetwith apreorder.A partialorder isareflexive,transitiveandantisymmetricrelation.A partiallyorderedset or poset isasetwithapartialorder.
Merl`esSubir`aCribillers
Proposition2.4. LetXbeafinitetopologicalspace.Thebinaryrelation ≤ onXdefinedbythefollowing expressionisapreorder: x ≤ yifandonlyifx ∈ Uy
Definition2.11. The Hassediagram ofaposet X isadirectedgraphwhoseverticesarethepointsof X and whoseedgesaretheorderedpairs(x , y )suchthat x < y andthereexistsno z ∈ X suchthat x < z < y Additionally,theelementsarearrangedindescendingorder,withbiggerelementsintheupperpartofthe diagram,whilesmalleronesareplacedbelow.
Example2.12. Let X = {a, b, c, d } withthefollowingopensets: ∅, {a, b, c, d }, {c}, {d }, {b, d }, {c, d } and {b, c, d },representedbytheinteriorsoftheclosedcurvesofFigure 1(a).
9 (2024),41–52;DOI:10.2436/20.2002.02.41.
Algebraictopologyoffinitetopologicalspaces
Since X is T0,itisaposet,thuswecantalkabouttheassociatedHassediagramof X .Let’sseehow itisconstructed.Westartwiththepointscorrespondingtoopensetsandplacethematthebottommost positions.Wecancomputetheopensets
= {c} andanalogously U
= {d }.With this,weestablishtheorderrelation: c < a,but c isnotcomparablewith
and d < b,but since b < a,wehavethechain
X .(b)Hassediagramof X
Nowlet’sseehowtoobtainatopologicalspacegivenatheHassediagramofaposet. Example2.13. Let Y betheposetgivenbytheHassediagramseeninFigure 2(a),wewanttocompute itsopensets.FollowingDefinition 2.6,wemovethroughtheHassediagramstartingfromthebottomand movingupwards.Thesets {c} and {d } areopen.
Now,consideranopenset U suchthat a ∈ U.Since c and d aresmallerthan a,theymustalsobe in U.Thus,wehavetheopenset Ua = {a, c, d }.Byfollowingasimilarprocessstartingfrom b,weobtain theopenset {b, c, d }.Whatwehavedonecanbedescribedas“placingourfinger”onthepoint a and descendingthroughallpossibleedgesuntilreachingthebottom.Itisimportantnottomissanyedges,for example, {b, d } isnotanopenset.Therefore,theopensetsof X are: {c}, {d }, {a, c, d }, {b, c, d },and theunions {c, d }, {a, b, c, d };seeFigure 2(b).
Definition3.1. Let x , y ∈ X betwopointsinafinitetopologicalspace.Wesaythat xcoversy if x > y andforall z ∈ X suchthat x > z ≥ y ,wehave z = y .Itcanalsobesaidthat yiscovered by x
Definition3.2. Let X beafinite T0 topologicalspace.Apoint x ∈ X iscalleda downbeatpoint ifit coversoneandonlyoneelementof X .Dually, x isan upbeatpoint ifitiscoveredbyexactlyoneelement. Pointsthatsatisfyeitherofthesepropertiesarereferredtoas beatpoints of X
Remark 3.3 IntheHassediagram, x isadownbeatpointifithasexactlyoneloweredge.Inthetopological space,thisisequivalenttosayingthattheset ˆ Ux = Ux \{x } hasamaximum.Similarly, x isanupbeat pointifithasexactlyoneupperedgeintheHassediagram.
WecanseeinExample 2.12 that b, d and c areupbeatpoints, b isalsoadownbeatpointand a is neitherofthem.TherearenobeatpointsinExample 2.13
Definition3.4. Afinite T0 topologicalspace X is minimal ifithasnobeatpoints.The core ofafinite topologicalspace X isasubspacethatisalsominimalasatopologicalspace.
Givenafinitetopologicalspace X ,itscorecanbeconstructedbyremovingbeatpointsoneata time.Thisprocesspreservesthehomotopypropertiesof X becausetheresultingsubspaceisastrong deformationretract(see[2,Proposition1.3.4]).Observethatthisminimalsubspacealwaysexists.If X has nobeatpoints,itisalreadyminimal,making X itsowncore,asillustratedbythespaceinExample 2.13.If beatpointsarepresent,theycanberemovedsuccessivelyuntilaminimalspaceisobtained.Forinstance, inExample 2.12,wecanretract d to b,then b to a,andlastly c to a;therefore, {a} isthecoreof X .As
(2024),41–52;DOI:10.2436/20.2002.02.41.
Algebraictopologyoffinitetopologicalspaces
youmayhavededuced,thespace X ishomotopictoapoint;therefore, X isindeedcontractible,which canbeeasilyobservedintheHassediagram,ratherthaninthedescriptionof X byitsopensets. Notethatminimalisinthesenseofnothavingbeatpoints,notofhavingafewpoints.Thiswillbe thesmallestsubspaceofafinitetopologicalspacethatkeepstheoriginalhomotopyproperties.Thespace ofExample 2.12 isnotminimal,whereastheoneinExample 2.13 isminimal.
Example3.5. Let X bethefinitetopologicalspaceassociatedwiththeHassediagramshowninFigure 3(a). Wecomputeitscorebyremovingthebeatpoints.First,since b isanupbeatpoint,weretractittowards a. Then,weretract c towards a becauseitisanupbeatpointof X \{b}.Finally,weretract e towards a becauseitisanupbeatpointof X \{b, c}.Theresultingsubspace X \{b, c, e} isminimalandtherefore isthecoreof X .Notethatchangingtheorderofthisprocessleadstothesameresult,bearinginmind thatwecanonlyretractonebeatpointatatime.
Example3.7. Considerthefollowingfinite T0 topologicalspaces X and Y givenbyFigure 4(a)and(c). Theyareverysimilar,butaretheyhomotopic?Wealreadycomputedthecoreof X inExample 3.5.For Y , wejustretract a,thatisadownbeatpoint,towards b andwehavethecoresshowninFigure 4(c)and(d).
X .(b)Thecoreof X .(c)Space Y .(d)Thecoreof Y
Figure3:(a)Space
Figure4:(a)Space
Merl`esSubir`aCribillers
Thecoresof X and Y arenothomeomorphic.Therefore,byStong’sTheorem 3.6, X and Y arenot homotopyequivalent.Butthesetwospacesarenotvisuallymuchdifferent,isthereanyotherwayto comparethem?Thenextsectionisdedicatedtorelaxingtheequivalencecriteriatoseethattheyaren’t thatdifferent.
4.McCord’sTheorem
McCordstudiesfinitetopologicalspacesbyassociatingthemwithabstractcombinatorialobjects,known as simplicialcomplexes.Thesecomplexescanbeassignedwithatopologicalrepresentation,calledthe geometricrealization,whichenablestheirvisualizationasgeometricstructuresinEuclideanspace,connected coherently,forminggeometricshapessuchastrianglesandtetrahedra.Ifthereaderisunfamiliarwiththese concepts,theycanrefertoanyintroductiontosimplicialcomplexes,suchas[5],[2,Appendix]or[7].
4.1Simplicialcomplexes
Definition4.1. A simplicialcomplex K consistsofaset VK,calledthesetofvertices,andaset SK of finitenonemptysubsetsof VK,whichiscalledthesetofsimplices,satisfyingthatanysubsetof VK of cardinalityoneisasimplexandanynonemptysubsetofasimplexisasimplex.
Definition4.2. The geometricrealization |K| ofasimplicialcomplex K isthesetofformalconvex combinations v ∈K αv v suchthat {v |αv > 0} isasimplexof K.
Definition4.3. Let X bea T0 finitetopologicalspace.The simplicialcomplexassociatedtoX ,or order complex,denotedby K(X ),isthesimplicialcomplexwhose n-simplicesarechainsoflength n:
x0 < x1 < < xn ,
wheretheorderrelationisgivenbyProposition 2.4
Example4.4. Let X bethefinite T0 topologicalrepresentedinFigure 5(a).Let’sseehowtoconstructthe associatedsimplicialcomplex K(X ).First,theelementsof X aretheverticesor0-simplices.Next,welook atthelongestchains.Herewehave {d < b < a} and {e < b < a}.Therefore,wehavetwo2-simplices. Notethattheyshareanedge, {b < a}
Next,wegodownindimension.Inthiscase,weneedtoaddanedgefrom c to d andanotherfrom c to e,becausetheotherchainsofsize1arealreadyrepresentedasedges(1-simplices)ofthe2-simplices. Finally,wegraphicallyrepresentthegeometricrealizationof K(X )inFigure 5(b).
Algebraictopologyoffinitetopologicalspaces
Theinverseprocesscanbedonebyconstructingthefaceposetof K,obtainingatopologicalspace. Thefaceposetisobtainedbytakingthesimplicesofagivensimplicialcomplex K aselementsanddefining theorderrelationbyinclusion.Notethatthiswouldnotresultintheoriginalspacebutinahomotopicone.
4.2TheMcCordmap
Definition4.5. Given K and L twosimplicialcomplexes,asimplicialmap φ : K→L isavertex map φ′ : VK → VL thatsendssimplicesintosimplices.
Definition4.6. Let X and Y befinite T0 topologicalspaces,and f : X → Y acontinuousmap.Then, the associatedsimplicialmap K(f ): K(X ) →K(Y )isdefinedby K(f )(x )= f (x ).
Example4.7. AspreviouslydiscussedinExample 3.5,nohomotopyequivalenceexistsbetween X and Y becausetheircoresarenothomeomorphic.Butusingthefollowingmap,wecanprovethatthereisaweak homotopyequivalencebetween X and Y .Consider X ′ and Y ′ thecoresof X and Y ,respectively.The map f : Y ′ → X ′ isgivenby f (
;seeFigure 6
Itisshownthat f isorder-preserving,andthereforecontinuous.Then,bythefactthatpreimagesof minimalopensetsarecontractible[7,Corollary2.2.12]and[4,Theorem6],weobtainthat f isaweak homotopyequivalence.
Everyhomotopyequivalenceinducesanisomorphismonthehomotopygroups,buttwospaceswith isomorphichomotopygroupsmaynotbehomotopyequivalent.Thiscorrespondenceiscalleda weak homotopyequivalence.Thismeansthatspaces X and Y haveisomorphichomotopygroups,andwiththis relaxedcriteria,wecanfinallysaythattheyareequivalent.Infact,theybotharefinitemodelsofthe sphere S 1
Observethat,givenafinite T0 topologicalspace X anditsgeometricrealization |K(X )|,anypoint α ∈ |K(X )| canbeexpressed,byconstruction,intermsofcoordinatesoverachain x1 < x2 < < xn in X in theform α = n i =1 λi xi ,where λi > 0forall1 ≤ i ≤ n and n i =1 λi =1.The support of α isprecisely thischain:supp(α)= {x1, x2,..., xn }
Figure6:Map f
Merl`esSubir`aCribillers
Definition4.8. Let X beafinite T0 topologicalspaceand α ∈|K(X )| apointinthegeometricrealization ofthesimplicialcomplexassociatedwith X suchthatsupp(α)= {x1, x2,..., xn }⊆ X .The McCordmap isthemap µX : |K(X )|→ X definedby µX (α)=min(supp(α))= x1
Example4.9. WestartfromExample 4.4,wherewecomputedthesimplicialcomplexassociatedwith atopologicalspace X .Apoint α ∈|K(X )| intheinteriorofthetriangle abe canbewrittenas α = λ1e + λ2b + λ3a with 3 i =1 λi =1.Then, µX (α)=min({e, b, a})= e,andthereforeeverypointinthe interiorofthistrianglewillmapto e,representedingreeninFigure 7.Similarly,pointsintheinterior of adb willmapto d ,representedinred.
Nowconsiderapointontheinterioroftheedge ae.Sincethesupportis {e, a},theminimumis e Moregenerally,thesmallestvertex,concerningtheposetorder,ofalltheonescontainedinthesimplex, will“absorb”thepointsintheinteriorofthesimplex.Figure 7 showstheMcCordmap,wherecolours representthepreimagesofthevertices.
Recallthatatriangulationisahomeomorphismbetweenatopologicalspaceandasimplicialcomplex. Forexample,atriangleishomeomorphicto S 1,orahollowtetrahedronishomeomorphicto S 2,andthese aresimplicialcomplexes.
Eachvertexislabelledwithanumber,andeachsimplexislabelledwiththenumbersofitsvertices. Wethenconstructthefaceposetof K andobtaintheHassediagramoftheassociatedposet;seeFigure 9. Notethatthisgivesusa minimalfinite topologicalspacethatmodelsthenon-finitespace S 2
Arichfamilyofsuchinvariantsfallsundertheumbrellaoftheso-calledBernstein–Satotheory,whose rootslieinthefoundationalworksofBernstein[2]andSato[23].Webrieflyoutlinetheirdiscovery.Denote by DR|C theringof C-lineardifferentialoperatorsonthepolynomialring R = C[x1,..., xn]andlet f bea nonzeropolynomial.Thenthereexistanonzerodifferentialoperator δ(s) ∈DR|C[s],andanon-constant monicpolynomial bf (s) ∈ C[s]satisfyingthefunctionalequation
Thepolynomial bf (s)istheBernstein–Satopolynomialof f
TheBernstein–Satopolynomialhasbeenthefocusofextensiveresearchsinceitencodesthebehavior ofthesingularitiesofthehypersurfacedefinedby f in Cn.Toshowcasethis,suppose f vanishesat0 ∈ Cn Awell-knowninvariantfromcomplexanalysisisthelog-canonicalthresholdof f attheorigin,definedas lct(f )=sup λ ∈ R>0 U 1 |f |2λ < ∞ forsomeneighborhood U oftheorigin
Thelog-canonicalthresholdisarationalnumberintheinterval(0,1].Themoresingular f is,thesmaller thelog-canonicalthresholdwillbe.Koll´arprovedthatthelog-canonicalthresholdof f isthesmallest rootof bf ( s)[13].Itisknownthattherootsof bf (s)arerationalandnegativeduetoMalgrangeand Kashiwara[16, 12].AnumberofinvariantshaveoriginatedaroundtheBernstein–Satopolynomialoverthe years.Ofspecialinterestinbirationalgeometryaremultiplieridealsandjumpingnumbers(forinstance, see[15]).
Tofixideas,let R bearegularringofcharacteristic p > 0and f anonzeroelement.Thetestideals (cf.Definition 2.10)areafamily {τ (f λ)}λ∈R≥0 ofidealsof R indexedbytherealnumbers.For λ ≤ µ, thesesatisfy τ (f λ) ⊇ τ (f µ),henceoneobtainsadescendingchainofidealsin R.Onecanshowthatfora fixed λ> 0,thereexists ε> 0suchthat τ (f λ)= τ (f µ)forall µ ∈ [λ, λ + ε),i.e.thefamilyisrightsemicontinuous.Onthecontrary,thereexistcertain λ> 0suchthat τ (I λ ε) ⊋ τ (I λ)forany ε> 0,thatis, thechainoftestideals“jumps”.Thesejumpingspotsarenamed F -jumpingnumbers(cf.Definition 2.14), andthesmallestamongthemisthe F -purethreshold,asintroducedin[24].Underfinitenesshypotheses, F -jumpingnumbersareknowntobediscreteandrational(seeTheorem3.1of[6]).Needlesstosay,these notionshavebeenextendedtonon-principalideals.
Astheterminologysuggests,thetestideals, F -jumpingnumbersand F -purethresholdsserveascharacteristic p > 0analoguestothemultiplierideals,jumpingnumbersandlog-canonicalthresholds,respectively. Remarkably,thereisadeepandintricaterelationshipbetweenthesetwotheories.Forinstance,onecan
PedroL´opezSancha
recoverthelog-canonicalthresholdfromthe F -purethresholdbyletting p →∞ (seeTheorem3.4in[19]). Itisalsoknowninseveralcasesthatthereductionmodulo p ofamultiplieridealproducesthecorresponding testideal[20].
The F -purethresholdhasbeencomputedinahandfulofcases.Itisknown,forinstance,inthecase ofellipticcurves,Calabi–Yauhypersurfaces,diagonalhypersurfacesanddeterminantalideals,tonamea few[3, 4, 10, 17].Amongthefewsituationswheretestidealshavebeenfullycharacterized,thereisthe caseofdeterminantalidealsofmaximalminors[9].
Ingeneral,finding F -jumpingnumbersandtestidealsisachallengingproblem,eveninsmoothambient spacessuchaspolynomialringsandwiththeaidofcomputationaltools.Tosomeextent,theaforementioned knownresultsrelyonthefavorablearithmeticandcombinatorialpropertiesoftheobjectsinvolved.Without theseproperties,verylittlecanbesaidabout F -invariants.
Ourgoalinthisarticleistocomputethe F -jumpingnumbersandtestidealsforanewclassof polynomials,whichwerefertoaslinearlysquare-freepolynomials.Thesearepolynomialswhosemonomials areallsquare-free,meaningtheyarenotdivisiblebyanysquareofanindeterminate.Intheprocess,wealso computeseveralother F -invariantsusefulforthetheory,namely,the ν-invariants,Frobeniusroots,and Bernstein–Satoroots,whichwewillintroduceinduecourse.Finally,werelatethesecomputationstothe log-canonicalthresholdoflinearlysquare-freepolynomialsincharacteristiczero.Thisworkoriginatedfrom thestudyof F -invariantsfordeterminantsofgenericmatricesofindeterminatesincharacteristic p > 0. Subsequently,itwasrealizedthatthesameideasappliedtolinearlysquare-freepolynomials.
Let R bearingofcharacteristic p > 0.Wedenoteby F : R → R, f → f p theFrobeniusor p-thpower map.Thisisaringendomorphismof R.Foraninteger e ≥ 0,welet F e : R → R, f → f pe bethe e-th iterateoftheFrobenius.
Definition2.1. Foraninteger e ≥ 0,the e-thFrobeniuspowerofanideal I ⊆ R is I [pe ] = F e (I )R =(f pe | f ∈ I ).
Thisisanidealof R.Inthecasethat I begeneratedby f1,..., fn,onehas I [pe ] =(
Remark 2.2 When I isaprincipalidealof R,say I =(f ),Frobeniuspowersandtheusualpowerscoincide, (f )[pe ] =(f )pe
Definition2.3. Foraninteger e ≥ 0,the e-thFrobeniusrootofanideal I ⊆ R isthesmallestideal J ⊆ R inthesenseofinclusionsuchthat
⊆ J [pe ]
Wedenotethe e-thFrobeniusrootoftheideal I by I [1/pe ].For e =0,weset I [1/pe ] = I
AcelebratedtheoremofKunzstatesthataring R ofcharacteristic p > 0isregularifandonlyifthe Frobenius F : R → R isaflatmap[14].Undertheassumptionofregularity,onecanshowthatFrobenius rootsarewell-defined.See,forinstance,Lemma2.3of[6].
Remark 2.4 Let I1, I2 beidealsof R suchthat I1 ⊆ I2.Thenonehas
Because I [1/pe ] 1 isthesmallestidealwith I1 ⊆ (I [1/pe
],itfollowsthat
henceFrobeniusrootspreserveidealcontainments.
Remark 2.5 Let I , J beidealsof R and e ≥ 0aninteger.Then
WenextdescribeanicecharacterizationofFrobeniusrootsintermsofgenerators,whichwillproveto becomputationallyuseful.Tothisend,weendow R withanexotic R-modulestructure.
Definition2.7. Foraninteger e ≥ 0,definethe R-module F e ∗ R asfollows.Itselementsaredenoted by F e ∗ f ,where f isin R.Asanabeliangroup, F e ∗ R isisomorphicto R,soadditionisdefinedby
Theactionof R on F e ∗ R isdefinedbyrestrictingscalarsalongthe e-thiterate F e oftheFrobenius,thatis,
),for r ∈
Definition2.8. ANoetherianring R ofcharacteristic p > 0isan F -finiteringif F e ∗ R isafinitelygenerated R-moduleforsome e ≥ 1(equivalently,all e ≥ 1).
Thenthee-thFrobeniusrootofIis I [1/pe ] =(gi ,j | i =1,..., m, j =1,..., n).
PedroL´opezSancha
2.2Testidealsand ν -invariants
Fromnowon,let R bearegular F -finiteringofcharacteristic p > 0.Forarealnumber x ∈ R,let ⌈x⌉∈ Z denotetheround-upof x,i.e.theleastintegergreaterorequalthan x
Asmentionedearlier,thetestidealsarethecharacteristic p > 0analoguesofthemultiplierideals.We adoptasadefinitionforthetestidealthecharacterizationgivenin[6]:
Definition2.10 ([6,Definition2.9]). Thetestidealofanideal I ⊆ R withexponent λ ∈ R≥0 is τ (I λ)= ∞ e=0 (I ⌈λpe ⌉)[1/pe ]
Remark 2.11 Itcanbeshownthattheidealsontheright-handsidegiveanascendingchainin R, (I
(seeLemma2.8in[6]).Since R isaNoetherianring,thechaineventuallystabilizes: τ (I λ)=(I ⌈λpe ⌉)[1/pe ],forsome e ≫ 0.
Remark 2.12 Let0 ≤ λ ≤ µ berealnumbers.Because ⌈λpe ⌉≤⌈µpe ⌉,onehasthat I ⌈λpe ⌉ ⊇ I ⌈µpe ⌉,forevery e ≥ 1.
F -jumpingnumberswereintroducedunderthename F -thresholdsin[19],asaninvarianttostudythe jumpingcoefficientsofthetestidealsofHaraandYoshida[8].Afterwards,itwasshownthatthesetsof F -thresholdsand F -jumpingnumbersareequal(see[6,Corollary2.30]).Onanothernote,onehasthe followingresultrelatingthelog-canonicalthresholdandthe F -purethreshold:
Anotherobjectcloselyrelatedtothe F -jumpingnumbersarethe ν-invariants:
Definition2.16 ([19]) Let I , J beidealsof R suchthat I ⊆ rad J,whererad J denotestheradicalof J Fixaninteger e ≥ 0.The ν-invariantoflevel e of I withrespectto J is
Because I ⊆ rad J,thisintegerexistsandisfinite.Theset ν• I (pe )of ν-invariantsoflevel e of I istheset ofintegersoftheform ν J I (pe )obtainedas J rangesovertheidealscontaining I initsradical:
• I (pe )= {ν J I (pe ) | J ⊆ R suchthat I ⊆ rad J}
Remark 2.17 InviewofProposition 2.6, r ≥ 0isthe ν-invariant ν J I (pe )ifandonlyif(I r )[1/pe ] ̸⊆ J
The ν-invariantswereintroducedpreciselytostudy F -thresholds.Infact,the F -threshold c J (I )of I withrespectto J wasdefinedin[19]as
Since F -thresholdsand F -jumpingnumberscoincidewhen R isaregularring,the ν-invariantsareapowerful toolforsheddinglightontestideals.
Incomputingthe ν-invariantsofanideal I ,itisnotevidenthowtochooseanideal J thatcontains I initsradical.Instead,however,onecaninspectthechainofideals
• I (pe )= {r ≥ 0 | (I r +1)[1/pe ] =(I r )[1/pe ]}
2.3Bernstein–Satoroots
ThelastalgebraicinvariantsrelevanttoourdiscussionaretheBernstein–Satoroots.Thesearecharacteristic p > 0analoguestotherootsoftheBernstein–Satopolynomialincharacteristiczero,aconcept
The p-adicvaluationon Z isthemap vp : Z → Z≥0 definedby vp (0)= ∞ and vp (n)=max {k ≥ 0 | p k divides n},for n =0, whichnaturallyextendstoavaluation vp : Q → Z≥0 byletting vp a b = vp (a) vp (b).
Thisinducesthe p-adicnorm |·|p : Q → R, |x|p = p vp (x ),andinturnthe p-adicmetric dp : Q × Q → R, dp (x, y )= p vp (x y ).Inthissetting,thering Qp of p-adicnumbersisthecompletionof Q withrespect tothe p-adicmetric.Thering Zp of p-adicintegersisthesubringof Qp givenby Zp = {α ∈ Qp ||
Because vp (n) ≥ 0forevery n ∈ Z,onehas |n|p ≤ 1,therefore Z iscontainedin Zp .Fromthedefinition, onealsoseesthat Q iscontainedin Qp .Asequence(xn)∞ n=0 ⊆ Q has p-adiclimit α ∈ Qp if xn → α in the p-adicmetric.Formoreon p-adicnumbers,werefertheinterestedreadertoSection7in[21]. Withthisinmind,Bernstein–Satorootsaredefinedasfollows:
Definition2.19 ([21,Proposition6.13],[22,TheoremIV.17]) Let I beanidealof R.A p-adicinteger α ∈ Zp isaBernstein–Satorootof I ifthereexistsasequence(νe )∞ e=0 ⊆ Z≥0 of ν-invariantsof I , νe ∈ ν• I (pe ), whose p-adiclimitis α
3.Linearlysquare-freepolynomials
Inthissectionweproveourmainresults,namely,thecomputationofBernstein–Satotheoryinvariantsfor linearlysquare-freepolynomialsincharacteristic p > 0.
Definition3.1. Let R = B[x1,..., xn]beapolynomialringoveracommutativering B.Wesaythata polynomialin R isalinearlysquare-freepolynomialifallitsmonomialsaresquare-free.
Example3.2. Let R = B[x11,..., x1n,..., xn1,..., xnn]beapolynomialringin n2 indeterminates.The indeterminatesmaybeassembledinan n × n genericmatrixofindeterminates X =(xij ).Thenthe determinantof X , det X = σ∈Sym(n) sgn(σ)x1σ(1) ··· xnσ(n), isalinearlysquare-freepolynomial.
Example3.3. Let X =(xij )bea2n × 2n skew-symmetricmatrixofindeterminates,thatis, xij = xji for1 ≤ i, j ≤ 2n.ThePfaffianof X isthepolynomial
ItcanbeshownthatthePfaffiansatisfies(Pf X )2 =det X .Sincenoindeterminateappearstwiceinthe samemonomial,thePfaffianislinearlysquare-free.
Example3.4. Let K beafieldand W ⊆ K E bearealizationofamatroid M,where E isafiniteset thatformsabasisof K E .Thentheconfigurationpolynomialof W islinearlysquare-free(see[7]).These polynomialshaveapplicationsinphysics.
Thepropositionbelowisawell-knownfactthatshowsthat F e ∗ R hasaparticularlynicestructure provided R isapolynomialringoveraperfectfieldofcharacteristic p > 0.Recallthatafield K of characteristic p > 0isperfectiftheFrobenius F : K → K isanautomorphismof K .Thisistantamount toeveryelementof K havinga pe -throotin K
Proposition3.5. LetR = K [x1,..., xn] beapolynomialringoveraperfectfieldKofcharacteristic Char(K )= p > 0.Foreachintegere ≥ 0,onehasthat
Inconsequence,theset {F e ∗
1
in n | 0
i1
in < pe } isabasisforF e ∗ R.Werefertothisasthe standardbasisofF e ∗ R.
WestartbycomputingtheFrobeniusrootsandthe ν-invariantsoflinearlysquare-freepolynomials. Thiswilllaythegroundworkforfurtherresults.Forthefollowinglemma,itwillbeconvenienttouse multi-indexnotation.If B[x1,..., xn]isapolynomialringin n variables,and a =(a1,..., an) ∈ Zn ≥0 isan n-tupleofnon-negativeintegers,welet
Lemma3.6. LetR = K [x1,..., xn] beapolynomialringoveraperfectfieldKofcharacteristicp > 0.Let fbealinearlysquare-freepolynomial.Fixanintegere ≥ 0.Thenforallintegers 0 ≤ r < pe ,F e ∗ f r isa nonzeroK-linearcombinationofelementsinthestandardbasisofF e ∗ R.
Proof. Because f islinearlysquare-free,onehas
forsomeinteger m ≥ 1,therefore
PedroL´opezSancha
Themonomialsintheexpressionabovehavetheform
Byassumption0 ≤ r < pe ,hencetheindeterminate xj appearsineachmonomialwithexponent
Itfollowsthat
,for i =1,...,
, isanelementinthestandardbasisof F e ∗ R.Asaresult,uptocollectingterms, F e ∗ f r reads
whichprovesthatthecoefficientsarein K .Because f r =0and F e ∗ R isafree R-module,somecoefficient isnonzero.
Theorem3.7. LetR = K [x1,..., xn] beapolynomialringoveraperfectfieldKofcharacteristicp > 0
Inthecase s =0,Lemma 3.6 showsthat F e ∗ f pe 1 isanonzero K -linearcombinationofelementsinthe standardbasisof F e ∗ R.ItfollowsfromProposition 2.9 thattheFrobeniusroot(f pe 1)[1/pe ] isgenerated byunitsof R,therefore(f pe 1)[1/pe ] = R.Nowsupposethat s ≥ 1.Inviewoftheascendingchainabove, toproveequalityitsufficestoverifythat (f )s ⊆ (f spe +pe 1)[1/pe ] and(f spe )[1/pe ] ⊆ (f )s
Proof. (i) Sincethesequence((pe 1)/pe )∞ e=0 haslimit1as e →∞,thereisaninteger e0 satisfying {λ}≤ (pe 1)/pe forall e ≥ e0.ItfollowsfromLemma 3.8 that(f ⌈λpe ⌉)[1/pe ] =(f )⌊λ⌋ for e ≥ e0, therefore τ (f λ)=(f )⌊λ⌋
(ii) Fixaninteger n ≥ 0.Then τ (f λ)=(f )⌊λ⌋ =(f )n forallrealnumbers λ with n ≤ λ< n +1.Onthe otherhand,onehas τ (f n+1)=(f )n+1.Consequently n +1isan F -jumpingnumberof f ,andtheassertion follows.
Proof. Let p > 0beaprimenumberand fp bethereductionmodulo p of f in Fp [x1,..., xn].If p does notdivideallthecoefficientsof f ,then fp isnonzeroandthuslinearlysquare-free,hencefpt(fp )=1by Lemma 3.8.Thisoccursforall p largeenough,solct(f )=1byTheorem 2.15
Proof. Let(td )∞ d =0 ⊆ Z≥0 beasequenceofnon-negativeintegersanddefine νd :=(td +1)p d 1,for d ≥ 0.
InviewofTheorem 3.7,each νd isa ν-invariantof f .Wethusobtainasequence(νd )∞ d =0 ⊆ Z≥0 of ν-invariantswith p-adiclimit νd → α = 1as d →∞.Inconsequence, α = fpt(f )isaBernstein–Sato rootof f .Becauseanysequenceof ν-invariantsof f isofthisform,itfollowsthat α =fpt(f )istheonly Bernstein–Satorootof f
Question3.12 ([21,Question6.16]) SupposethattheF-purethreshold α ofanidealIliesin Z(p),the localizationof Z at {pk | k ≥ 0}.IsthelargestBernstein–SatorootofIequalto α?
Theanswerisaffirmativeforlinearlysquare-freepolynomialsinanycharacteristic p > 0.
Thegoalofthisarticleistofillthepictureintroducedinthissection.InSection 2 werecallthefundamentalsofsymplecticandPoissongeometryfromtheHamiltonianformalismofmechanics.InSection 3 we introduceLiealgebroidsand E -symplecticmanifoldsasobjectsbetweensymplecticandPoissonstructures. Wewilladditionallygiveexamplesofinterestwheretheyhavebeenfruitfullyapplied.InSection 4 we remembertheinterplaybetweenconservedquantitiesandsymmetries,codifiedinthemomentmapofa Hamiltonianaction.Thepresenceofsymmetriesallowsforeliminationofdegreesoffreedom,aprocedure formalizedbythereductiontheoremofMarsdenandWeinstein.WepresentSternberg’sandWeinstein’s constructions,andshowhowtheyhavebeenextendedtothesettingof E -symplecticmanifolds.
coordinates pi , qi ,canberecoveredfromafunction H ∈C∞(R2n),calledtheHamiltonian,following Hamilton’sequationsofmotion:
.(1)
Theprevioussetofequationscanbecompactlywrittenusingmatrixnotationas XH = J ·∇f ,where J is thestandardskew-symmetricmatrix.CommonchoicesinphysicsfortheHamiltonianareenergyfunctions oftheform H = 1 2m p2 i + V (q)forsomesmoothfunction V ∈C∞(R2n),calledthepotentialofthe system.
Poissonbracketswereoriginallyintroducedtostudytheevolutionofobservables,i.e.,smoothfunctions, alongtheHamiltoniandynamics.Inmoremathematicalterms,ifwedefinethePoissonbracketof H and f tobethederivativeof f alongtheflowof XH ,Hamilton’sequations(1)directlyshow
Eventhoughanybracketarisingfrom(3)satisfiestheseconditions,therearebracketsfulfillingthese propertieswhichcannotbedefinedinthisway.AtrivialexampleisthePoissonbracket {f , g } =0for all f , g ∈C∞(M).Thesystematicstudyoftheseobjectsisthebranchof Poissongeometry.
Definition2.3. A Poissonbracket onasmoothmanifold M isabilinear,skew-symmetricoperation {·, ·} : C∞(M) ×C∞(M) →C∞(M)satisfyingLeibniz’sruleineachargumentandJacobi’sidentity.
Definition3.1. A Liealgebroid isavectorbundle π : A→ M togetherwithavectorbundlemap ρ : A→ TM coveringtheidentityandequippedwithaLiebracket[ , ]A onthespaceofsectionsΓA.Moreover, thebracketsatisfies,forany X , Y ∈ ΓA and f ∈C∞(M),thefollowingcompatibilityconditions:
Inequation(5a),theoperator L denotestheLiederivativeofafunctionalongavectorfield.
Letustakeabriefdetourandpreciselydescribehowtheseobjectsariseinthedescriptionofsystems withsingularities.Wewillpresentexamplesarisingfromphysicswhereallthefollowingassumptionsare satisfied.Considerthattheequationsofmotionofoursystemcanbedescribedintermsofa C∞ subsheafofvectorfields F⊆ X.Furthermore,assumethesheafislocallyfinitelygenerated,thatis,forany point p ∈ M thereisanopenset U containing p andsections X1,..., Xm ∈FU suchthattheirrestriction toanyopenset V ⊆ U generates FV .Wecanmaketwoadditionalassumptions,eachofwhichgivesrise towell-knownobjectsindifferentialgeometry.
• Ifweadditionallyassumetheintegrabilitycondition[F , F ]= F ,thesheaf F definesa singular foliation inthesenseofAndroulidakisandSkandalis.Theseobjectscanbeintegratedtogivestandard singularfoliations,orfoliationsinthesenseofStefanandSussman.
Apromenadethroughsingularsymplecticgeometry
• Ifthesheafisnotonlylocallyfinitelygeneratedbutalsolocallyfree,itisatheoremofSerre[11]in thealgebraicsettingandSwan[13]inthecontinuouscaseshowsthesheaf F canberecoveredas thesheafofsectionsofavectorbundle E
Ifbothassumptionsaresimultaneouslymade,weget projectivefoliations or Debordfoliations.If E isa representingvectorbundlefor F inthesensethat F =ΓE ,wegetanaturalmapofvectorbundles ρ : E → TM givenbytheevaluationofasectionatapoint.Thismapiscalledtheanchor,andisinjectiveinan openanddensesubset U ⊆ M,i.e.,genericallyinjective.Theintegrabilitycondition[F , F ] ⊆F liftstoa bracketinthespaceofsectionsΓE .Onecaneasilycheckthatthecompatibilityconditions(5a)and(5b) aresatisfiedand,thus,anysuchobjectisaninstanceofaLiealgebroid.
Definition3.2. Let M beasmoothmanifold.An E-structure isthechoiceofaDebordfoliation F⊆ X or,equivalently,avectorbundle π : E → M withagenericallyinjectivemap ρ : E → TM.Wecallthe pair(E , M)an E-manifold
Inthepreviousformula,anargumentwithahatimpliesithasbeensuppressedfromthecollectionof inputs.Aroutineverificationshowsd2 A =0.Thecohomologyspacesofthecomplexof A-formsarecalled Liealgebroidcohomologygroups. Withthenotionofdifferentialformsandexteriorcalculusforsystemswithconstraints,wecandefine asymplecticformmimickingthestandarddefinitioninclassicaldifferentialgeometry.
Definition3.3. Let π : A→ M beaLiealgebroid.A symplecticform on A isatwoform ω ∈ Ω2 A(M)which isclosedandnon-degenerate.Wecallthepair(A, ω)a symplecticLiealgebroid.Similarly,if π : E → M isan E -manifold,wecallthepair(E , ω)an E-symplecticmanifold
Thereareobstructionsforthemap µ• : g →C∞(M)tobeamorphismofLiealgebras,whereweendow C∞(M)withthePoissonbracketasLiebracket.ThefailuretohaveaLiealgebramorphismismeasured bythemap
c(X , Y )= {µ X , µ Y }− µ[X ,Y ]
Astheprojectionofthiselementto XHam(M)vanishesbythecommutativityofthediagram,wecanidentify theimage c(X , Y )withanelementinthekernelker X• ≃ R.Thismapthusdeterminesanelementin theChevalley–Eilenbergcomplex, c ∈ C(g; R).Themap c isclosed,andhencedeterminesaclassinthe cohomologygroupH2(g; R).Thelift µ• canbechosentobeaLiealgebramorphismifandonlyif[c]=0. Moreover,allpossiblesuchchoicesareparametrizedbyelementsofthegroupH1(g; R).
AssumingsomeconditionsontheLiegroup G , 2 thereisauniquelydeterminedliftwhichwecallthe comomentmap.Byconstruction,itintertwinestheadjointactionin g withtheinducedpullbackaction in C∞(M).Asthenamehints,however,itisbettertothinkofthecomomentmapintermsofadual objectcalledthe momentmap
Definition4.1. Let(M, ω)beasymplecticmanifoldandlet G beaLiegroupactingon M.Wesaythe actionis Hamiltonian ifthereexistsamap µ : M → g∗,calledthe momentmap,satisfyingthefollowing conditions:
Wecallanysuchtriple(M, ω, µ)a HamiltonianG-space
Equation(7a)isreminiscentoftheequivarianceofthecomomentmap.Equation(7b)isadirect consequenceofthefactthatthefundamentalvectorfieldsof g actinaHamiltonianfashion:indeed,all wearesayingisthat ⟨µ, X ⟩ istheHamiltonianfunctionof X # or,following(6),thecomomentmap µX
LetusassumenowwearegivenaHamiltonian G -space(M, ω, µ)witha G -invariantHamiltonian H andequationsofmotiondeterminedby XH .Assume H isaninvariantfunctionundertheactionof G .We willalsoassumethegroup G isconnected.ThistechnicalconditionensuresthatanyHamiltonianaction isalsoasymplecticaction,thatis,preservestheform ω.Twoconsequencesarisefromthesefacts.
Firstly,observe XH is G -invariant.Werecallthesymplecticform ω is G -invariantbecauseweassume G isconnected.BecausetheHamiltonian H is G -invariant,wehave
Asaconsequence,wededuce XH = ρg · XH
Secondly,assumewearegivenaregularvalue α ∈ g∗ of µ.Byequivariance,everyothervalue β ∈Oα isregular,andthusthepreimage Mα = µ 1(Oα)isasubmanifold.Byinvarianceof H wehave LX # H =0. Thedefinitionofmomentmapimpliesnow
2Forexample,if G issemisimple,weknowbyWhitehead’slemmasthatH1(g; R)=H2(g; R)=0.Thesemisimplicity assumptionisautomaticallysatisfiedif G isacompactgroup.
PabloNicol´asMart´ınez
Asthisequalityholdsforall X ∈ g,wehave ⟨dµ, XH ⟩ =0.Thisresultimplies XH istangenttothelevel setsof µ and,by G -invariance,tothepreimagesofthecoadjointorbits.
Thesymplectomorphisminducedbythischoiceofconnectioniscalledthe minimalcoupling ofthe system.GivenaHamiltonianinthebasespace, H ∈C∞(T∗M),wecanconsideritspullbacktoeither spaceandgetequivalentdynamics.Theinducedequationsofmotionarecalled Wong’sequations [8].
3Namely,freenessandproperness.Thelatterissatisfiedif G isacompactgroup,whichwehaveenforcedsincethe constructionofthemomentmap.
4Inthisformula,themomentmap µP :T∗P → g ∗ isthenaturalmomentmapobtainedforthecotangentliftofany G -actiontothecotangentbundlewiththecanonicalsymplecticform.
Throughouttherestofthesectionwewillfixan E -manifold EM → M andaprincipal G -bundle P → M TheproofofTheorem 4.4 followsessentiallythesameargumentasWeinstein.Thecomplicationliesin developingthemachinerynecessarytostateandfollowtheoriginalproof.Sincewewouldliketoextend thesingularitiesofourconfigurationspace M tothebundles EM and P,weneedaproceduretodoso.The fundamentalnotionisthatof prolongation,whichdatesbackatleasttotheworksofdeLe´on,Marrero, andMart´ınez[1].
Definition4.5. Assume f : N → M isasurjectivesubmersionoveraLiealgebroid A→ M.The prolongation of A along f ,written Lf A,isthepullbackbundleofthemorphismsdf :TN → TM and ρ : A→ M Asaset,itcanbeidentifiedwith
Throughouttherestofthesection,wefixan E -manifold E → M.Theprolongationofthedual bundle E ∗ M → M,whichcanbethoughtasthesingulartangentbundleofthecotangentbundle,carries anaturalLiouvilleformwhosedifferentialissymplectic[1].Thus,theprolongationof E ∗ M isasymplectic E -manifold,instrongresemblancetothecotangentbundleofasmoothmanifold.
PabloNicol´asMart´ınez
Similarly,weconsidertheprolongationoftheprincipal G -bundle P → M.Because EP → P hasa naturalactionoftheLiegroup G andtheanchormapisinjectiveonanopenanddensesubset,theaction on P liftstoanactionon EP whichfactorsthroughthestandardtangentmap.Byduality,theactionlifts tothedualbundle E ∗ P and,moreover,itbecomesHamiltonianwithrespecttothecanonicalsymplectic structure.ThefactthattheactionoftheLiegroup G automaticallyliftsto EP isonlyvalidfor E -manifolds. IfwewanttoestablishsimilarresultsforsymplecticLiealgebroids,strongercompatibilityassumptionsare neededtodefineHamiltoniangroupactions(see[3]).
Let f :(Cn+1,0) → (C,0)beagermofaholomorphicfunction(orequivalentlyaconvergentpower series f ∈ C{x0,..., xn})withanisolatedsingularityattheorigin.UsingthecanonicalmixedHodge structureofthecohomologygroupsoftheMilnorfiberof f ,Steenbrink[7]definedthe Hodgespectrum of f asthegeneratingfunction
isthe Milnornumber andthepositiverationalnumbers
formadiscretesetofinvariantsofthesingularity f called Hodgespectralexponents (or spectralnumbers). Theyaresymmetricwithrespectto(n +1)/2,i.e.,forevery j =1,..., µ,wehave αµ+1 j =(n +1) αj andthusitisenoughtostudythemintheinterval(0,(n +1)/2].
AnotherinterestingfeatureprovedbyVarchenko[10]isthattheHodgespectralexponentsof f are stableunderdeformationswithconstantMilnornumber µ.Adeformationofahypersurface f (x0,..., xn) ∈ C{x0,..., xn} isafamilyofhypersurfaces ft1 ,...,tk (x0,..., xn)forsomesetofparameters(t1,..., tk ) ∈ S ⊆ Ck , satisfying f (x0,..., xn)= f0,...,0(x0,..., xn).Then,inVarchenko’sresultweareaskingthattheMilnornumber of ft1 ,...,tk (x0,..., xn)isthesameforall(t1,..., tk ) ∈ S
Question1. Forwhichlimitsofsequencesofhypersurfaces (f (i ))i ⩾0 doesthedistributionofHodgespectral exponentsDf (i ) (s) convergetoNn+1(s)?Equivalently,forwhichlimitsof (f (i ))i ⩾0 doesthecharacteristic function χf (i ) (T ) convergeto F{Nn+1}(τ )= T 1 log T n+1?
Here, pg denotesthe geometricgenus of f definedas pg =dimC(R n 1 π∗OX )0 for n ⩾ 2(pg =dimC(π∗OX /OC2 )0 for n =1), with π : X → Cn+1 beingaresolutionofthesingularity.M.Saito[5]provedarelationbetweenthisinvariant andtheHodgespectralexponents,namely pg =#{i | αi ⩽ 1},andthusDurfee’sconjecturepredictsthat 1isadominatingvaluefor n =2.K.Saitoaskedwhetheronecangeneralizethisstatement:
Let f :(C2,0) → (C,0)beagermofaholomorphicfunction,orequivalentlyaconvergentpower series f ∈ C{x, y }.Theequation f =0defineslocallya(complex)planecurvearoundtheorigin.We onlyconsiderirreducibleplanecurves f (alsocalledplanebranches),i.e.,irreducibleelementsoftheunique factorizationdomain C{x, y }
Theorem2.1 (Puiseux) Letf ∈ C{x, y } defineanirreducibleplanecurvethatisnottangenttothey-axis (i.e., ∂f ∂x =0).ThenthereisaPuiseuxseriess (x)= i ⩾0 ai x i /m suchthatf (x, s(x))=0.Moreover, allsuchseriesareconjugates σε(s)= i ⩾0
i x i /m with εm =1.Thecurvecanbeparameterized byt → t m , i ⩾0 ai t i
APuiseuxseriesof f hastheform s(x)= j ∈(e0
with e0 = m,
i =min{j | aj
/
(ei 1)},
i =gcd(ei 1, βi )(i =1,..., g ), where m ischosensuchthat eg =1.Since ei |ei 1,wecandefine ni = ei 1/ei ⩾ 2.
Thesenumericalinvariantshaveageometricmeaning: e0 isthemultiplicityof f attheoriginand ei (i =1,..., g )isthemultiplicityof f atthe i -thrupturedivisorofitsminimalembeddedresolution,or equivalentlythelastinfinitelynearpointofthe i -thclusterofconsecutivesatellitepoints.Theseconcepts areexplainedin[2].
FromthePuiseuxserieswecandefine:
Definition2.2. The characteristicexponents ofaplanebranch f aretherationalnumbers β1 m ,...,
(i =1,..., g ) with nj ⩾ 2, lj ⩾ 1,gcd(lj , nj )=1.Fromthiswedefine:
Definition2.3. The Puiseuxpairs ofanirreducibleplanecurve f are(n1, l1),...,(ng , lg ).
ThecharacteristicexponentsandthePuiseuxpairsaretwoequivalentsetsofcompletetopological invariantsofthesingularityof f .Thatis:theydetermine,andaredeterminedby,thehomeomorphismclass of f 1(0) ∩ U forasmallenoughneighbourhood U oftheorigin.
Remark 2.4. ThenamePuiseuxpairsappearinvariousslightlydifferentwaysintheliterature.Webasedour definitionontheonegivenbyM.Saito[6],whousedthisnameforthepairs(k1, n1),...,(kg , ng )with k1 = n1 + l1, ki = li .Casas-Alvero[2]usedthesimilartermcharacteristicpairstoreferto(β1, m),...,(βg , m).
InthissectionwestudythecaseofplanebranchesforK.Saito’sQuestion 1 onthelimitdistributionofHodgespectralexponents.Weconsiderirreducibleplanecurves f ∈ C{x, y } withPuiseuxpairs (n1, l1),...,(ng , lg ).Inthiscase,K.Saito’sQuestion 1 asksforwhichlimitsofirreducibleplanecurves f doesthedistributionofHodgespectralexponents Df (s)convergeto N2(s)(recallDefinition 1.2).Equivalently,itasksforwhichlimitsofthePuiseuxpairs(n1, l1),...,(ng , lg )doesthecharacteristicfunction χf (T )= F{Df (s)}(τ )convergetotheFouriertransform F{N2(s)}(τ )= T 1 log T 2
K.Saitocomputedtheparticularcaseofalimitwheretheinvariant ng ofthelastPuiseuxpairtends toinfinitywhilealltheremainingPuiseuxpairsarekeptfixed.Hisresult,recallingProposition 1.6,isthat theresultinglimitdistributionofHodgespectralexponentsis lim ng →+∞ χf (T )= T 1
Inthethesis,westudyfromseveralpointsofviewtheso-calledstochasticdifferentialequationsdrivenby afractionalBrownianmotionwithHurstindex H > 1/2.Theseobjectsaredifferentialequationsofthe form
where B H isafractionalBrownianmotionwith H ∈ (1/2,1),thatis,acenteredstochasticGaussian processwithcovariancefunction
Noticethat,inparticular,when H =1/2then B H isastandardBrownianmotion.Thefirsttopiccovered inthethesisisgivingsensetoanequationlike(1).Thefactthat B H isnotasemimartingalemakes itimpossibletodefinetheintegralwithrespectto B H inasimilarwayasitisdefinedforthestandard Brownianmotion.InvirtueoftheresultsofYoungin[4]andthefurthercontributionsofZ¨ahlein[5],we areabletodefinetheintegralwithrespectto B H inthegeneralizedStieltjessense.Oncethestochastic integraliswell-defined,wefollowcloselytheargumentsofNualartandR˘a¸scanuin[2]toprovetheexistence anduniquenessofsolutionstoageneralSDEoftheform(1).
where σ isdeterministic, σ and b satisfythesamehypothesisasfortheexistenceofthedensityfunction pt (x)andweassumefurtherthatthereexist0 <λ< Λsuchthat λ<σs < Λ,thenthedensity pt (x) for t ∈ (0, T ]isboundedinthefollowingway:
EnaquesttreballestudiemlaregularitatdelesfrontereslliuresLipschitzenel problemad’Alt–Caffarelli.DemostremquelesfrontereslliuresLipschitzs´on C 1,α mitjan¸cantlainvari`anciaperreescalamentdelproblemailaregularitatinicial Lipschitzdelafrontera.Am´esam´es,tamb´eprovemquelesfronteres C 1,α s´on C ∞,cosaque,juntamentambelresultatanterior,implicaquelesfronteres lliuresLipschitzs´on C ∞
Theproofofthistheoremisaccomplishedintwosteps:first,byprovingthatLipschitzfreeboundaries are C 1,α,andsecond,byshowingthat C 1,α freeboundariesaresmooth.Forthefirststep,themain ideaistousetherescalinginvarianceof(1).Noticethatif u isasolution,thenforany r > 0the function ur (x)= 1 r u(rx)isalsoasolutioninthecorrespondingrescaleddomain B1/r .Thispropertyis crucialtofinishthisfirststep.Geometrically,theLipschitzregularityof ∂{u > 0}∩ B1 impliesthatthefree boundaryalwaysremainsoutsideaconeofafixedopening.Usingthatoursolutionsatisfies(1),weare abletoshowthatwecanimprovetheopeningofthisconeintheball B1/2.However,thisaloneisclearly insufficienttoconcludethatthefreeboundaryis C 1,α.Whatenablesustocompletetheproofofthisfirst stepispreciselytherescalinginvarianceoftheproblem,whichweusetorepeattheopeningimprovement iterativelyinthesequenceofballs B2 k .Intuitively,thisprocesstellsusthatthefreeboundary“flattens” aswezoominattheoriginwhichimplies,aftersomeextrasteps,thatthefreeboundaryis C 1,α
Asforthesecondstep,weperformsomecomputationscombinedwithSchauderestimatesforthe Laplaciantoshowthatoncewehave C 1,α regularityonthefreeboundary,thenwecanimprovethe boundaryasmuchaswewanttoobtain C ∞ regularity.Lastly,combiningbothstepsandusingasimple coveringargumentweobtaintheproofofthetheorem.
Recentstudieshavedescribedthegenerationandcontrolofmovementsintermsofthebenefitsand costsassociatedwithpotentialmovements[7],thusestablishingafundamentalrelationshipbetweenmovementgenerationanddecision-makingtheory.Thecombinationofmovementrelatedchoicesandmore cognitivedecisionsdeterminesourresponsesandthebehaviourwithwhichweinteractwiththeenvironment.Thiscanbestudiedandmodelledmathematicallythrough optimaldecisionmaking and motor controltheory [8].However,thesetheoriesfallshorttoconsiderthecontributionandroleoftheinner perceptionofourbody,namelythebodilyperceptionor proprioception,whichplaysacrucialrolewhen planningandexecutingmovements.Inparticular,proprioceptionprovidesinternalcorroborationthata movementisongoing,itishenceadistributedphenomenonimplicatedinprocessesof top-downprediction and bottom-upcorrection
Uncosdenombres K ´esmonogensielseuanelld’entersest`ageneratperun solelementcoma Z-`algebra.Enelcasc´ubic,determinarsi K ´esmonogenono ´esequivalentaresoldrel’equaci´odiof`antica |IK (X , Y )| =1sobre Z,on IK ´esla forma´ındexdelcos.Unasoluci´oenteradeterminaunpuntracionalalacorbade g`enere1 IK (X , Y )= Z 3.Mitjan¸cantaquestaconstrucci´o,espotdemostrarque K determinauna F3-`orbitaen H 1(Q, E [3]),on E ´eslacorbael l´ıpticadefinida per Y 2 =4X 3 +Disc(K ).Donemlaconstrucci´oexpl´ıcitad’aquesta`orbitapelcas decossosc´ubicspursicaracteritzemlasumadecociclesassociatsacossosno isomorfs.
Itiswellknown,duetotheprimitiveelementtheorem,thatanynumberfield K isgeneratedbyasingle algebraicnumberover Q.Onewouldthinkthattheanalogousstatementshouldholdfortheringof integers OK ,sothat OK = Z[α]forsomealgebraicinteger α ∈OK .However,Dedekindfoundthefirst counterexampleforthisassumptionin1878(see[2]).Whenthereexistssuch α, K issaidtobe monogenic Today,for n ≥ 3,itisexpectedthat,whenorderedbydiscriminant,thesetofmonogenicnumberfieldsof degree n hasmeasure0(see[1]).
Afterchoosingasuitableintegralbasisof OK ,wecanassociateadegree n(n 1)/2homogeneous form IK on n 1variablesto K calledthe indexform of K .Thisformallowstocharacterizethemonogenity of K byadiophantineequation: K ismonogenicif,andonlyif,thereexist x1,..., xn 1 ∈ Z suchthat IK (x1,..., xn 1)= ±1.When n =3, IK isabinarycubichomogeneousformanditsdiscriminantisequal tothediscriminantof K (see[3]).Thus,theprojectivecurve CK : IK (X , Y )= Z 3 issmoothandan integralsolutiontotheindexformequationgivesrisetoarationalpointon CK .Therefore,wefocuson studyingtheexistenceofrationalpointson CK
Fornon-zero r ∈ Q,let E r denotetheellipticcurvegivenby Y 2 =4X 3 + r .Let D =Disc(K ).In recentworkofAlp¨oge,Bhargava,andShnidman(see[1]),theydefinedarationalmap πK : CK → E 27D anda3-isogeny ϕD : E D → E 27D suchthat(CK , πK )isa ϕD -coveringof E 27D .Asaconsequence, CK isahomogeneousspacefor E D .The ϕD -coveringsof E 27D areparametrizedby H 1(Q, E D [ϕD ]),
where E D [ϕD ]=ker(ϕD ),andhomogeneousspacesfor E D areparametrizedbytheWeil–Chˆatelet group H 1(Q, E D ),whosetrivialclassconsistsofthehomogeneousspacesfor E D whichhavearational point.ThesecohomologygroupsarerelatedbytheKummerexactsequence
where ι isgiven,intermsof ϕD -coveringsandhomogeneousspaces,by ι(CK , πK )= CK .Thus,(CK , πK ) isinthekernelof ι if,andonlyif, CK hasarationalpoint.Therefore,byanalysingthiskernel,wecan studythemonogenityoffamiliesofnumberfieldswithdiscriminant D.Weapplythistheoryinorderto giveboundsforthetotalnumberofmonogeniccubicnumberfieldswiththesamediscriminantinterms of E D
Since(CK , πK )isa ϕD -covering,itdeterminesaclass αK ∈ H 1(Q, E D [ϕD ]).When K isaDedekind typeIfield,i.e.when K = Q( 3 √hk 2),where h, k arecoprime,square-freeintegerssuchthat hk 2 ̸≡±1 modulo9,weprovethatthecocycle
Solvingnonlinearequationsoftheform f (x)=0isacommonchallengeinvariousscientificfields,spanningfrombiologytoengineering.Whenalgebraicmanipulationisnotfeasible,iterativemethodsbecome necessarytodeterminesolutions.Newton’smethodisawell-knownapproach,derivedfromlinearizingthe equation f (x)=0.Itsiterativeexpressionisgivenby:
where yn = xn f (xn ) f ′(xn ) isaNewton’sstepand δ isthedampingparameter.Noticethat δ =0corresponds toNewton’smethodand δ =1toTraub’smethod.Newton’smethodconvergesquadraticallyforsimple
